Recent zbMATH articles in MSC 16https://www.zbmath.org/atom/cc/162022-05-16T20:40:13.078697ZWerkzeugBasic abstract algebra. Exercises and solutionshttps://www.zbmath.org/1483.000032022-05-16T20:40:13.078697Z"Mortad, Mohammed Hichem"https://www.zbmath.org/authors/?q=ai:mortad.mohammed-hichemPublisher's description: This book is mainly intended for first-year University students who undertake a basic abstract algebra course, as well as instructors. It contains the basic notions of abstract algebra through solved exercises as well as a ``True or False'' section in each chapter. Each chapter also contains an essential background section, which makes the book easier to use.Maximal submodule graph of a modulehttps://www.zbmath.org/1483.050752022-05-16T20:40:13.078697Z"Alwan, Ahmed H."https://www.zbmath.org/authors/?q=ai:alwan.ahmed-hSummary: Let \(U\) be a left \(R\)-module where \(R\) is a commutative ring with identity. The maximal submodule graph \(MG (U)\) of proper \(R\)-submodules of \(U\) is an undirected graph defined as follows: the vertex set is the set of all proper \(R\)-submodules of \(U\), and there is an edge between two distinct vertices \(N\) and \(L\) if and only if \(N + L\) is a maximal submodule of \(U\). We study these graphs to relate the combinatorial properties of \(MG (U)\) to the algebraic properties of the \(R\)-module \(U\). We study connectedness for \(MG (U)\). We investigated some properties of \(MG (U)\) such as, diameter, girth, and clique number.Some properties of intersection graph of a module with an application of the graph of \(\mathbb{Z}_n\)https://www.zbmath.org/1483.050792022-05-16T20:40:13.078697Z"Orhan Ertaş, Nil"https://www.zbmath.org/authors/?q=ai:orhan-ertas.nil"Sürül, Sema"https://www.zbmath.org/authors/?q=ai:surul.semaSummary: Let \(R\) be an unital ring which is not necessarily commutative. The intersection graph of ideals of \(R\) is a graph with the vertex set which contains proper ideals of \(R\) and distinct two vertices \(I\) and \(J\) are adjacent if and only if \(I \cap J \neq 0\) is denoted by \(G\). In this paper, we will give some properties of regular graph, triangle-free graph and clique number of \(G(M)\) for a module \(M\). We also characterize girth of an Artinian module with connected module. We characterize the chromatic number of \(\mathcal{G}(\mathbb{Z}_n)\). We also give an algorithm for the chromatic number of \(\mathcal{G}(\mathbb{Z}_n)\).The Tate module of a simple abelian variety of type IVhttps://www.zbmath.org/1483.111212022-05-16T20:40:13.078697Z"Banaszak, Grzegorz"https://www.zbmath.org/authors/?q=ai:banaszak.grzegorz"Kaim-Garnek, Aleksandra"https://www.zbmath.org/authors/?q=ai:kaim-garnek.aleksandraSummary: The aim of this paper is to investigate the Galois module structure of the Tate module of an abelian variety defined over a number field. We focus on simple abelian varieties of type IV in Albert classification. We describe explicitly the decomposition of the \(\mathcal{O}_\lambda [G_F]\)-module \(T_\lambda (A)\) into components that are rationally and residually irreducible. Moreover these components are non-degenerate, hermitian modules that rationally and residually are non-degenerate, hermitian vector spaces.Kloosterman sums over finite Frobenius ringshttps://www.zbmath.org/1483.111702022-05-16T20:40:13.078697Z"Nica, Bogdan"https://www.zbmath.org/authors/?q=ai:nica.bogdanSummary: We study Kloosterman sums in a generalized ring-theoretic context, that of finite commutative Frobenius rings. We prove a number of identities for twisted Kloosterman sums, loosely clustered around moment computations. Our main result is the computation of the fourth moment of twisted Kloosterman sums, for non-primitive twists.Matrices over noncommutative rings as sums of \(k\)th powershttps://www.zbmath.org/1483.112152022-05-16T20:40:13.078697Z"Katre, S. A."https://www.zbmath.org/authors/?q=ai:katre.shashikant-a"Wadikar, Kshipra"https://www.zbmath.org/authors/?q=ai:wadikar.kshipraSummary: The first author and \textit{A. S. Garge} [Proc. Am. Math. Soc. 141, No. 1, 103--113 (2013; Zbl 1276.11160)] obtained necessary and sufficient trace conditions for matrices over commutative rings with unity to be sums of \(k\)th powers. In this paper, we show that these conditions hold for matrices over noncommutative rings too. For \(n \geq 2\), we deduce Vaserstein's result for sums of squares of matrices and also obtain nice trace conditions for \(n \times n\) matrices to be sums of cubes. For an integer \(n \geq k \geq 2\), we get a sufficient condition \(\pmod k\) for an \(n \times n\) matrix over a noncommutative ring to be a sum of \(k\)th powers.Some remarks on Kida's formula when \(\mu\neq 0\)https://www.zbmath.org/1483.112442022-05-16T20:40:13.078697Z"Lim, Meng Fai"https://www.zbmath.org/authors/?q=ai:lim.meng-faiSummary: Kida's formula in classical Iwasawa theory relates the Iwasawa \(\lambda\)-invariants of \(p\)-extensions of number fields. Analogue of this formula was subsequently established for the Iwasawa \(\lambda\)-invariants of Selmer groups under an appropriate \(\mu=0\) assumption. In this paper, we give a conceptual (but conjectural) explanation that such a formula should also hold when \(\mu\neq 0\). The conjectural component comes from the so-called \(\mathfrak{M}_H(G)\)-conjecture in noncommutative Iwasawa theory.Cayley graphs versus algebraic graphshttps://www.zbmath.org/1483.130112022-05-16T20:40:13.078697Z"Pranjali, Pranjali"https://www.zbmath.org/authors/?q=ai:pranjali.pranjali"Kumar, Amit"https://www.zbmath.org/authors/?q=ai:kumar.amit-n|kumar.amit.2|kumar.amit|kumar.amit.1"Yadav, Tanuja"https://www.zbmath.org/authors/?q=ai:yadav.tanujaAuthors' abstract: Let \(\Gamma\) be a finite group and let \(S\subseteq \Gamma\) be a subset. The Cayley graph, denoted by \(\mathrm{Cay}(\Gamma, S)\) has vertex set \(\Gamma\) and two distinct vertices \(x,y \in \Gamma\) are joined by a directed edge from \(x\) to \(y\) if and only if there exists \(s\in S\) such that \(x = sy\). In this manuscript, we characterize the generating sets \(S\) for which \(\mathrm{Cay}(\Gamma, S)\) is isomorphic to some algebraic graphs, namely, unit graphs, co-unit graphs, total graph and co-total graphs.
Reviewer: Nitin Bisht (Indore)Grassmannians and cluster structureshttps://www.zbmath.org/1483.130372022-05-16T20:40:13.078697Z"Baur, Karin"https://www.zbmath.org/authors/?q=ai:baur.karinGrassmannians are classical mathematical structures that appear in the study of Lie theory, algebraic geometry, combinatorics and many other areas. In these notes Baur introduces Grassmannians and their interplay with cluster theory.
The notes have few prerequisites for the first sections, so a student with a good course in linear algebra and some notions of commutative algebra can follow them with no problem. The introduction contains several references and exercises. This proposes a hands-on approach to the reader.
By the end of Section 1, the reader can understand the definition of cluster algebra of type A and the relation between cluster algebras and Postnikov diagrams, that is the essence of J. Scott's key theorem in the area.
In Section 2 the author introduces quiver with potentials arising from dimer models and the boundary algebra, which is the link to study categorification for cluster algebras in the sense of Jensen-King-Su.
By Section 3 the author gives a primer on her latest research on the topic [\textit{K. Baur} et al., Proc. Lond. Math. Soc. (3) 113, No. 2, 213--260 (2016; Zbl 1386.13060); Nagoya Math. J. 240, 322--354 (2020; Zbl 1452.05187); Algebra Number Theory 15, No. 1, 29--68 (2021; Zbl 1459.05346)] while still providing several examples and references. This part is more suitable for graduate students and researchers interested in cluster algebras arising from Grassmannians.
Reviewer: Ana Garcia Elsener (Buenos Aires)On Rayner structureshttps://www.zbmath.org/1483.130392022-05-16T20:40:13.078697Z"Krapp, Lothar Sebastian"https://www.zbmath.org/authors/?q=ai:krapp.lothar-sebastian"Kuhlmann, Salma"https://www.zbmath.org/authors/?q=ai:kuhlmann.salma"Serra, Michele"https://www.zbmath.org/authors/?q=ai:serra.micheleThe article ``On Rayner structures'' by Lothar Sebastian Krapp, Salma Kuhlmann and Michele Serra explores the algebraic and combinatorial properties of generalised power series fields. More specifically, given the field \(k((G))\) of \(k\)-valued power series in a totally ordered abelian group \(G\) (which can be realised as the space of k-valued functions on \(G\) with well ordered support), and a set \(\mathcal{F}\) of well ordered subsets of \(G\), the article explores the \emph{\(k\)-hull} of \(\mathcal{F}\), and establishes necessary conditions for this to satisfy appropriate algebraic properties.
The paper begins by giving a list of algebraic and set theoretic properties (Conditions 2.1) labelled (S1)--(S6), (A1)--(A5) that can be satisfied by the set \(\mathcal{F}\), and recalls from a previous work of Rayner that the \(k\)-hull \(k((\mathcal{F}))\) of \(\mathcal{F}\) is a subfield of \(k((G))\) in the event that \(\mathcal{F}\) satisfies an appropriate collection of these conditions. Explicitly, \(k((\mathcal{F}))\) is an additive subgroup when it satisfies conditions (S2), (S3), and (S5), it is a subring when it also satisfies (A3) and (A4), and it is a subfield when it also satisfies (A1). However, these are merely sufficient conditions, and the aim of the paper is to establish necessary conditions for these properties to hold.
The first result of the article, Proposition 3.4, states that provided \(k\neq\mathbb{F}_2\), \(k((\mathcal{F}))\) is indeed an additive subgroup of \(k((G))\) if and only if \(\mathcal{F}\) is closed under taking unions and subsets, and contains the singleton {0}, i.e. if and only if \(\mathcal{F}\) satisfies (S2), (S3) and (S5). This strengthens Rayner's original result (Theorem 3.1(i)), showing that the original condition is indeed necessary. The result does not hold if \(k=\mathbb{F}_2\), as demonstrated by Example 3.6.
The authors go on to demonstrate a necessary and sufficient condition for \(k((\mathcal{F}))\) to be a subring of \(k((G))\) in Proposition 3.9, provided k is a sufficiently large field. This result gives a stronger condition than that originally stated by Rayner, since it only requires \(\mathcal{F}\) to satisfy (S2), (S3), (S5) and (A2), while (A3) and (A4) are unnecessary.
The remainder of the paper focuses on field structure. The aim is to find necessary and sufficient conditions for \(k((\mathcal{F}))\) to be a field, a \emph{Hahn field} and a \emph{Rayner field}. Briefly, a Hahn field is a subfield of \(k((G))\) containing all polynomials, and a Rayner field is a \(k\)-hull \(k((\mathcal{F}))\) where \(\mathcal{F}\) satisfies (S2), (S3), (S5), (A1), (A3) and (A4), which is a subfield of \(k((G))\) by Rayner's original theorem.
The final main results of the paper are Proposition 3.15 and Theorem 3.18. Proposition 3.15 gives a necessary and sufficient condition for \(k((\mathcal{F}))\) to be a subfield and a Hahn field in terms of the Conditions 2.1, at least in the case where \(k\) has characteristic 0. Theorem 3.18 states that if \(k((\mathcal{F}))\) is a Raynor field then it is a Hahn field, and that the converse holds when \(k\) has characteristic 0. The article concludes by stating that the \(k\)-hull \(k((\mathcal{F}))\) is a Rayner field if and only if it is a Hahn field, if and only if it satisfies all of Conditions 2.1.
Overall, this paper should be of interest to anyone who is concerned with power series and generalisations thereof, but since it is a short article with very understandable proofs, I would say that it is accessible to anyone algebraically minded, and certainly worth reading.
Reviewer: Adam Jones (Manchester)The probability when a finite commutative ring is nil-cleanhttps://www.zbmath.org/1483.130402022-05-16T20:40:13.078697Z"Danchev, P."https://www.zbmath.org/authors/?q=ai:danchev.peter-vassilev"Samiei, M."https://www.zbmath.org/authors/?q=ai:samiei.mahdiSummary: We define an indicator of the probability when a finite commutative ring is nil-clean, and calculate this probability for certain classes of finite commutative rings.Compact connected components in relative character varieties of punctured sphereshttps://www.zbmath.org/1483.140172022-05-16T20:40:13.078697Z"Tholozan, Nicolas"https://www.zbmath.org/authors/?q=ai:tholozan.nicolas"Toulisse, Jérémy"https://www.zbmath.org/authors/?q=ai:toulisse.jeremyGiven an oriented surface \(\Sigma_{g,s}\) of genus \(g\) with \(s\) punctures, and a non-compact semi-simple Lie group \(G\), let \(\Gamma_{g,s}:=\pi_1(\Sigma_{g,s})\). The character variety is defined by the GIT quotient \(\mathfrak{X}(\Sigma_{g,s},G):=\mathrm{Hom}(\Gamma_{g,s},G)/\!/G\), where \(G\) acts on \(\mathrm{Hom}(\Gamma_{g,s},G)\) by conjugation.
For \(s>0\), \(\mathfrak{X}(\Sigma_{g,s},G)\) does not reflect all the geometric properties of \(\Sigma_{g,s}\), namely one could have \(\mathfrak{X}(\Sigma_{g,s},G)\simeq\mathfrak{X}(\Sigma_{g',s'},G)\) as long as \(\chi(\Sigma_{g,s})=\chi(\Sigma_{g',s'})\), where \(\chi(\Sigma_{g,s})\) is the Euler characteristic of \(\Sigma_{g,s}\). The relative character varieties in some sense overcome this obstacle. Denoted by \(c_1,\dots, c_s\) the homotopy classes of loops going counter-clockwise around the \(s\) punctures, for an element \(h=(h_1,\dots,h_s)\in G^s\), define \(\mathrm{Hom}_h(\Gamma_{g,s},G):=\{\rho: \Gamma_{g,s}\to G\ |\ \rho(c_i)\in C(h_i)\}\), here \(C(h_i)\) is the conjugacy class of \(h_i\) in \(G\). The associated relative variety is thus \(\mathfrak{X}_h(\Sigma_{g,s},G):=\mathrm{Hom}_h(\Gamma_{g,s},G)/\!/G\).
In the present paper, the authors study the relative character varieties \(\mathfrak{X}_h(\Sigma_{0,s},G)\) for the classical Hermitian Lie groups \(\mathrm{SU}(p,q), \mathrm{Sp}(2n,\mathbb{R})\) and \(\mathrm{SO}^*(2n)\). By applying the nonabelian Hodge correspondence between relative character varieties and moduli spaces of parabolic Higgs bundles, the authors prove the main result of the paper: when \(s\geq3\), there exists \(h\) such that \(\mathfrak{X}_h(\Sigma_{0,s},G)\) has a compact connected component (when the Higgs fields are nilpotent) which contains a Zariski dense representation.
Reviewer: Pengfei Huang (Heidelberg)Nash blowups in prime characteristichttps://www.zbmath.org/1483.140242022-05-16T20:40:13.078697Z"Duarte, Daniel"https://www.zbmath.org/authors/?q=ai:duarte.daniel-c-s"Núñez-Betancourt, Luis"https://www.zbmath.org/authors/?q=ai:nunez-betancourt.luisSummary: We initiate the study of Nash blowups in prime characteristic. First, we show that a normal variety is non-singular if and only if its Nash blowup is an isomorphism, extending a theorem by \textit{A. Nobile} [Pac. J. Math. 60, No. 1, 297--305 (1975; Zbl 0324.32012)]. We also study higher Nash blowups, as defined by T. Yasuda. Specifically, we give a characteristic-free proof of a higher version of Nobile's theorem for quotient varieties and hypersurfaces. We also prove a weaker version for \(F\)-pure varieties.The hulls of matrix-product codes over commutative rings and applicationshttps://www.zbmath.org/1483.140472022-05-16T20:40:13.078697Z"Deajim, Abdulaziz"https://www.zbmath.org/authors/?q=ai:deajim.abdulaziz"Bouye, Mohamed"https://www.zbmath.org/authors/?q=ai:bouye.mohamed"Guenda, Kenza"https://www.zbmath.org/authors/?q=ai:guenda.kenzaThis paper explores the hull of the matrix product codes over a commutative ring. For a matrix product code \([\mathcal{C}_1, \cdots, \mathcal{C}_s]A\) where \(\mathcal{C}_{1}, \cdots, \mathcal{C}_s\) and linear codes of the same length and A a given matrix, some connections between the hull of \([\mathcal{C}_{1}, \cdots, \mathcal{C}_s]A\) and the hull of \(\mathcal{C}_{1}, \cdots, \mathcal{C}_s\) are provided. This allowed to give some necessary and sufficient conditions for a matrix-product codes to be LCD. Under some assumptions, LCD matrix product codes over the residue field of a finite chain are provided by using torsion codes and the existence of asymptotically good sequences of LCD matrix product codes over finite chain ring is proved.
Reviewer: Joël Kabore (Ouagadougou)Rank varieties and \(\pi \)-points for elementary supergroup schemeshttps://www.zbmath.org/1483.140902022-05-16T20:40:13.078697Z"Benson, Dave"https://www.zbmath.org/authors/?q=ai:benson.david-john"Iyengar, Srikanth B."https://www.zbmath.org/authors/?q=ai:iyengar.srikanth-b"Krause, Henning"https://www.zbmath.org/authors/?q=ai:krause.henning"Pevtsova, Julia"https://www.zbmath.org/authors/?q=ai:pevtsova.juliaThis paper is about developing \(\pi\)-points for the elementary supergroup schemes. Let \(k\) be a field of positive characteristic \(p\geq 3\). In \(k[t,\tau]\), let \(t\) has even and \(\tau\) has odd degrees. Consider the algebra
\[
A := \dfrac{k[t,\tau]}{(t^p-\tau^p)}.
\]
Let also \(E\) to be an elemntary supergroup scheme over \(k\). Denote by \(kE\) the dual \(\mathrm{Hom}_k(k[E],k)\). If \(M\) is any \(kE\)-module, and \(K\) be the extension field of \(k\), then define the following \(kE\)-modules,
\[
M_k:= K\otimes_k M, \quad M^k:= \mathrm{Hom}_{k}(K,M).
\]
These two can be also defined as
\[
M_k:= KE\otimes_{kE} M, \quad M^k:= \mathrm{Hom}_{kE}(KE,M).
\]
Definition. A \(\pi\)-point of \(E\) consists of an extension field \(K\) of \(k\) together with a map of graded \(K\)-algebras \(\alpha:A_K\rightarrow KE_K\), for \(A_K:=K\otimes_k A,\) such that \(KE_K\) has finite flat dimension as an \(A_K\)-module via \(\alpha\).
Let \(P = k[[\boldsymbol{s},\sigma]]\), where \(\boldsymbol{s}=s_1,\dots, s_n\) and \(I\) be the ideal generated by
\[
s_1^p,\dots, s_{n-1}^p,s_n^{p^m},s_n^p-\sigma.
\]
Then if \(J:=(s_1^p,\dots, s_{n-1}^p,s_n^{p^m})\subset k[[\boldsymbol{s}]]\), then the map \(\alpha:A\rightarrow kE\) is determined by polynomials \(f(\boldsymbol{s})\) and \(g(\boldsymbol{s})\) in \(k[\boldsymbol{s}]\) uniquely defined such that
\[
f(\boldsymbol{s})^p\equiv g(\boldsymbol{s})^2s_n^p\quad \text{modulo} \, J,
\]
where \(\alpha(t)=f(\boldsymbol{s})\) and \(\alpha(\tau)=g(\boldsymbol{s})\sigma\). Now set \(P_\alpha:= \dfrac{k[[\boldsymbol{s},\sigma]]}{(f(\boldsymbol{s})^p-g(\boldsymbol{s})^2\sigma^2)}\). In this case, \(\alpha\) can be factored as
\[
A \overset{\dot{\alpha}}{\rightarrow}P_\alpha\overset{\bar{\alpha}}{\rightarrow}kE,
\]
whre \(\dot{\alpha}:(t,\tau)\mapsto(f(\boldsymbol{s}),g(\boldsymbol{s})\sigma)\), and \(\bar{\alpha}\) is the canonical surjection.
The following theorem says when a map \(A\rightarrow kE\) can be a \(\pi\)-point.
Theorem 1. Let \(\alpha\) does not factor through the surjection \(A\rightarrow k\). Then the map \(\dot{\alpha}\) has finite flat dimension, and for any \(kE\)-module M the module \(\alpha^*(M)\) has finite flat dimension if and only if \(\bar{\alpha}^*(M)\) has finite flat dimension.
It is also shown that every \(kE\)-module \(M\) is projective if and only if \(\alpha^*(M_K)\), or equivalently \(\alpha^*(M^K)\), has finite flat dimension for every \(\pi\)-point \(\alpha:A_K\rightarrow KE_K\).
Let now \(\alpha:A_K\rightarrow KE_K\) and \(\beta:A_L\rightarrow KE_L\) are two \(\pi\)-points. we say they are equivalent, and write \(\alpha \sim\beta\), if for each finitely generated \(kE\)-module \(M\), the \(A_K\)module \(\alpha^*(M_K)\) has finite flat dimension if and only if the \(A_L\)-module \(\beta^*(M_L)\) has finite flat dimension. The following theorem expresses other ways to say \(\alpha \sim\beta\).
Theorem 2. Let \(E\) be a supergroup scheme and \(\alpha:A_K\rightarrow KE_K\) and \(\beta:A_L\rightarrow KE_L\) are two \(\pi\)-points of \(E\). Then the followings are equivalent:
1) \(\alpha\) and \(\beta\) are equivalent.
2) For all \(kE\)-modules \(M\), the \(A_K\)-module \(\alpha^*(M_K)\) has finite flat dimension if and only if the \(A_L\)-module \(\beta^*(M_L)\) has finite flat dimension.
3) For all \(kE\)-modules \(M\), the \(A_K\)-module \(\alpha^*(M^K)\) has finite flat dimension if and only if the \(A_L\)-module \(\beta^*(M^L)\) has finite flat dimension.
4) \(\mathfrak{p}(\alpha) = \mathfrak{p}(\beta).\)
Thus the map sending \(\alpha\) to \(\mathfrak{p}(\alpha)\) induces a bijection between the set of equivalence classes of \(\pi\)-points and the set \(\mathrm{Proj}H^{*,*}(E, k)\).
Definition. The \(\pi\)-support of a \(kE\)-module, \(M\), which is denoted by \(\pi-\mathrm{supp}_E(M)\), is the subset of \(\mathrm{Proj}H^*(E,k)\) consisting of primes \(\mathfrak{p}(\alpha)\) where \(\alpha\) is a \(\pi\)-point such that the flat dimnesion of \(\alpha^*(M_K)\) is finitie. Replacing \(\alpha^*(M_K)\) by \(\alpha^*(M^K)\) we have the \(\pi\)-cosupport of \(M\), denoted by \(\pi-\mathrm{cosupp}_E(M)\).
It is shown that, with the notation as above, \(M\) is projective if and only if \(\pi-\mathrm{supp}_E(M) = \emptyset\), or equivalently \(\pi-\mathrm{cosupp}_E(M)=\emptyset\).
Then the support and cosupport of the \(kE\)-module \(M\) are defined by
\[
\mathrm{supp}_E(M):=\{\mathfrak{p}\in \mathrm{Proj}H^{*,*}(E,k)|\Gamma_{\mathfrak{p}}(M)\neq 0\},
\]
\[
\mathrm{cosupp}_E(M):=\{\mathfrak{p}\in \mathrm{Proj}H^{*,*}(E,k)|\Lambda^{\mathfrak{p}}(M)\neq 0\}.
\]
For an elementary supergroup scheme \(E\) and the \(kE\)-module \(M\), it is also shown that \(\mathrm{supp}_E(M)=\pi\mathrm{supp}_E(M)\) and \(\mathrm{cosupp}_E(M)=\pi-\mathrm{cosupp}_E(M)\).
Theorem 3. Let \(E\) be an elementary supergroup scheme over \(k\). The stable module category \(\mathrm{StMod}(kE)\) is stratified as a \(\mathbb{Z}/2\)-graded triangulated category by the natural action of the cohomology ring \(H^{*,*}(E,k)\). Therefore, the assignment
\[
C\mapsto \bigcup_{M\in C} \mathrm{supp}_E(M)
\]
gives a one to one correspondence between the localizing subcategories of \(\mathrm{StMod}(kE)\) invariant under the parity change operator \(\Pi\) and the subsets of \(\mathrm{Proj}H^{*,*}(E,k)\).
Reviewer: Fereshteh Bahadorykhalily (Shiraz)Representations of cohomological Hall algebras and Donaldson-Thomas theory with classical structure groupshttps://www.zbmath.org/1483.141032022-05-16T20:40:13.078697Z"Young, Matthew B."https://www.zbmath.org/authors/?q=ai:young.matthew-bCohomological Hall algebras of quivers with potential were introduced by Kontsevich and Soibelman as tractable algebraic analogues of Donaldson-Thomas invariants of Calabi-Yau threefolds, and as a mathematical model of ``BPS algebras'' studied in theoretical physics by Harvey and Moore. In this context, representations of cohomological Hall algebras are expected to correspond to ``open'' BPS invariants in a purely algebraic setting.
The present paper studies a class of representations of cohomological Hall algebras attached to quivers with involution and equivariant potential. In the geometric incarnation, this corresponds to an extension of Donaldson-Thomas theory from structure group \(GL(n,\mathbb C)\) to the other classical groups. This is expected to model ``orientifold'' BPS invariants. The main result of the paper is an extension of the integrality results of Kontevich-Soibelman and Reineke to this equivariant setting.
The paper fits into the author's larger program on the mathematical underpinnings and repercussions of the orientifold construction.
Reviewer: Johannes Walcher (Montréal)On the structure of ternary Clifford algebras and their irreducible representationshttps://www.zbmath.org/1483.150152022-05-16T20:40:13.078697Z"Abłamowicz, Rafał"https://www.zbmath.org/authors/?q=ai:ablamowicz.rafalThe author studies the structure of the algebra generated over \(\mathbb{C}\) by generators \(e_1,\dots,e_n\) whose third powers are equal to 1 and \(e_k e_\ell=\omega e_\ell e_k\) for each \(k<\ell\) where \(\omega=e^{\frac{2\pi i}{3}}\). He concludes that when \(n\) is even, this algebra is isomorphic to \(M_{\frac{n}{2}}(\mathbb{C})\), and when it is odd, it is isomorphic the direct sum of three complex matrix algebras of degree \(\frac{n-1}{2}\).
Reviewer's note: The results reported in the paper appeared earlier in a much greater generality, see for instance [\textit{M. Vela}, Commun. Algebra 30, No. 4, 1995--2021 (2002; Zbl 1011.16030)].
Reviewer: Adam Chapman (Tel Hai)Noncommutative polynomial algebras of solvable type and their modules. Basic constructive-computational theory and methodshttps://www.zbmath.org/1483.160012022-05-16T20:40:13.078697Z"Li, Huishi"https://www.zbmath.org/authors/?q=ai:li.huishiPublisher's description: Noncommutative Polynomial Algebras of Solvable Type and Their Modules is the first book to systematically introduce the basic constructive-computational theory and methods developed for investigating solvable polynomial algebras and their modules. In doing so, this book covers:
\begin{itemize}
\item A constructive introduction to solvable polynomial algebras and Gröbner basis theory for left ideals of solvable polynomial algebras and submodules of free modules
\item The new filtered-graded techniques combined with the determination of the existence of graded monomial orderings
\item The elimination theory and methods (for left ideals and submodules of free modules) combining the Gröbner basis techniques with the use of Gelfand-Kirillov dimension, and the construction of different kinds of elimination orderings
\item The computational construction of finite free resolutions (including computation of syzygies, construction of different kinds of finite minimal free resolutions based on computation of different kinds of minimal generating sets), etc.
\end{itemize}
This book is perfectly suited to researchers and postgraduates researching noncommutative computational algebra and would also be an ideal resource for teaching an advanced lecture course.\(D3\)-modules versus \(D4\)-modules -- applications to quivershttps://www.zbmath.org/1483.160022022-05-16T20:40:13.078697Z"D'Este, Gabriella"https://www.zbmath.org/authors/?q=ai:deste.gabriella"Keskin Tütüncü, Derya"https://www.zbmath.org/authors/?q=ai:keskin-tutuncu.derya"Tribak, Rachid"https://www.zbmath.org/authors/?q=ai:tribak.rachidSummary: A module \(M\) is called a \(D4\)-module if, whenever \(A\) and \(B\) are submodules of \(M\) with \(M = A \oplus B\) and \(f: A \rightarrow B\) is a homomorphism with \(\operatorname{Im}f\) a direct summand of \(B\), then \(\operatorname{Ker}f\) is a direct summand of \(A\). The class of \(D4\)-modules contains the class of \(D3\)-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain \(R\), for an \(R\)-module \(M\) which is a direct sum of cyclic submodules, \(M\) is direct projective (equivalently, it is semi-projective) iff \(M\) is \(D3\) iff \(M\) is \(D4\). Also we prove that, over a prime PI-ring, for a divisible \(R\)-module \(X\), \(X\) is direct projective (equivalently, it is Rickart) iff \(X \oplus X\) is \(D4\). We determine some \(D3\)-modules and \(D4\)-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on \(D3\)-modules and \(D4\)-modules via quivers.On a class of \(g\)-lifting moduleshttps://www.zbmath.org/1483.160032022-05-16T20:40:13.078697Z"Ghawi, Thaar Younis"https://www.zbmath.org/authors/?q=ai:ghawi.thaar-younisSummary: The aim of this note is to introduce and investigate a category of modules which is analogous to that of \(g\)-lifting and principally lifting modules. The module \(M\) is called principally \(g\)-lifting if for every cyclic submodule \(K\) of \(M\) there exists a decomposition \(M = A \oplus B\) such that \(A\) is a submodule of \(K\) and \(K \cap B\) is \(g\)-small in \(M\). As an application, we defined principally \(g\)-semiperfect modules and characterize these modules in terms of principally \(g\)-lifting modules.Structures related to right duo factor ringshttps://www.zbmath.org/1483.160042022-05-16T20:40:13.078697Z"Chen, Hongying"https://www.zbmath.org/authors/?q=ai:chen.hongying"Lee, Yang"https://www.zbmath.org/authors/?q=ai:lee.yang"Piao, Zhelin"https://www.zbmath.org/authors/?q=ai:piao.zhelinSummary: We study the structure of rings whose factor rings modulo nonzero proper ideals are right duo; such rings are called \textit{right FD}. We first see that this new ring property is not left-right symmetric. We prove for a non-prime right FD ring \(R\) that \(R\) is a subdirect product of subdirectly irreducible right FD rings; and that \(R / N_\ast (R)\) is a subdirect product of right duo domains, and \(R / J ( R )\) is a subdirect product of division rings, where \(N_\ast( R )\) (\textit{J(R)}) is the prime (Jacobson) radical of \(R\). We study the relation among right FD rings, division rings, commutative rings, right duo rings and simple rings, in relation to matrix rings, polynomial rings and direct products. We prove that if a ring \(R\) is right FD and \(0 \neq e^2 = e \in R\) then \textit{eRe} is also right FD, examining that the class of right FD rings is not closed under subrings.Annihilating property of zero-divisorshttps://www.zbmath.org/1483.160052022-05-16T20:40:13.078697Z"Jung, Da Woon"https://www.zbmath.org/authors/?q=ai:jung.dawoon"Lee, Chang Ik"https://www.zbmath.org/authors/?q=ai:lee.chang-ik"Lee, Yang"https://www.zbmath.org/authors/?q=ai:lee.yang"Nam, Sang Bok"https://www.zbmath.org/authors/?q=ai:nam.sang-bok"Ryu, Sung Ju"https://www.zbmath.org/authors/?q=ai:ryu.sung-ju"Sung, Hyo Jin"https://www.zbmath.org/authors/?q=ai:sung.hyo-jin"Yun, Sang Jo"https://www.zbmath.org/authors/?q=ai:yun.sang-joBy the authors, in this paper, a ring is called right AP if every nonzero right annihilator of an element contains a nonzero ideal. Properties of right AP rings are studied. For example, it is shown that a ring \(R\) is right AP if and only if \(D_n(R)\) is right AP for every integer \(n\) such that \(n\geq 2\), where \(D_n(R)\) is the ring of \(n\times n\) upper triangular matrices over \(R\) whose diagonal elements are equal.
Reviewer: Jae Keol Park (Pusan)Rings of polynomials with Artinian coefficientshttps://www.zbmath.org/1483.160062022-05-16T20:40:13.078697Z"Johnson, F. E. A."https://www.zbmath.org/authors/?q=ai:johnson.francis-e-aSummary: We study the extent to which the weak Euclidean and stably free cancellation properties hold for rings of Laurent polynomials \(A[t_1,t_1^{-1},\ldots,t_n,t_n^{-1}]\) with coefficients in an Artinian ring \(A\).Arithmetical ringshttps://www.zbmath.org/1483.160072022-05-16T20:40:13.078697Z"Tuganbaev, A. A."https://www.zbmath.org/authors/?q=ai:tuganbaev.askar-aThis long paper is a survey, with some new results, on arithmetical rings, modules, and Bezout rings (not necessarily commutative).
The plan of this article is very clear. There is a table of contents which is very useful for the reader. This paper contains 5 chapters. Each chapter starts with the wording of its main results and then the materials necessary to prove these results are introduced. For instance the main results of Chapter 3, ``Rings with flat and quasi-projective ideals'' are:
``3A. Theorem [the author, Usp. Mat. Nauk 35, No. 5(215), 245--246 (1980; Zbl 0451.16018); Math. Notes 38, 631--636 (1985; Zbl 0601.16004); translation from Mat. Zametki 38, No. 2, 218--228 (1985); J. Math. Sci., New York 213, No. 2, 268--271 (2016; Zbl 1338.13033); translation from Fundam. Prikl. Mat. 19, No. 2, 207--211 (2014)]). For an invariant semiprime ring \(A\), the following conditions are equivalent:
\begin{itemize}
\item[(1)] \(A\) is an arithmetical ring;
\item[(2)] every submodule of any flat \(A\)-module is a flat module;
\item[(3)] every finitely generated ideal of the ring \(A\) is a quasi-projective right \(A\)-module.
\end{itemize}
3B. Theorem [\textit{C. U. Jensen}, Proc. Am. Math. Soc. 15, 951--954 (1964; Zbl 0135.07902)]. A commutative ring \(A\) is an arithmetical semiprime ring if and only if every submodule of any flat \(A\)-module is a flat module.
3C. Theorem [\textit{A. A. Tuganbaev}, J. Math. Sci., New York 213, No. 2, 268--271 (2016; Zbl 1338.13033); translation from Fundam. Prikl. Mat. 19, No. 2, 207--211 (2014)]. If \(A\) is an invariant ring, then \(A\) is an arithmetical ring if and only if every its finitely generated ideal is a quasi-projective right \(A\)-module such that all endomorphisms can be extended to endomorphisms of the module \(A_A\).''
The sequel of Chapter 3 is devoted to the study of flat modules and the proofs of these theorems.
This very interesting paper will be very useful and a reference for each mathematician working on these subjects.
Reviewer: François Couchot (Caen)Socle deformations of selfinjective orbit algebras of tilted typehttps://www.zbmath.org/1483.160082022-05-16T20:40:13.078697Z"Skowroński, Andrzej"https://www.zbmath.org/authors/?q=ai:skowronski.andrzej"Yamagata, Kunio"https://www.zbmath.org/authors/?q=ai:yamagata.kunioSummary: We survey recent development of the study of finite-dimensional selfinjective algebras over a field which are socle equivalent to selfinjective orbit algebras of tilted type. Main aim is to present a characterization of these selfinjective algebras from [the authors, Frobenius algebras. II: Tilted and Hochschild extension algebras. Zürich: European Mathematical Society (EMS) (2017; Zbl 1378.16001)] and, as an application, give a new characterization of selfinjective Nakayama algebras socle equivalent to selfinjective orbit algebras of tilted type.
For the entire collection see [Zbl 1461.16002].Weak \(FI\)-extending modules with ACC or DCC on essential submoduleshttps://www.zbmath.org/1483.160092022-05-16T20:40:13.078697Z"Tercan, Adnan"https://www.zbmath.org/authors/?q=ai:tercan.adnan"Yaşar, Ramazan"https://www.zbmath.org/authors/?q=ai:yasar.ramazanSummary: In this paper we study modules with the \(WFI^+\)-extending property. We prove that if \(M\) satisfies the \(WFI^+\)-extending, pseudo duo properties and \(M\)/(Soc \(M)\) has finite uniform dimension then \(M\) decompose into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if \(M\) satisfies the \(WFI^+\)-extending, pseudo duo properties and ascending chain (respectively, descending chain) condition on essential submodules then \(M=M_1 \oplus M_2\) for some semisimple submodule \(M_1\) and Noetherian (respectively, Artinian) submodule \(M_2\). Moreover, we show that if \(M\) is a \(WFI\)-extending module with pseudo duo, \(C_2\) and essential socle then the quotient ring of its endomorphism ring with Jacobson radical is a (von Neumann) regular ring. We provide several examples which illustrate our results.On weakly prime and weakly 2-absorbing modules over non-commutative ringshttps://www.zbmath.org/1483.160102022-05-16T20:40:13.078697Z"Groenewald, Nico J."https://www.zbmath.org/authors/?q=ai:groenewald.nico-johannesSummary: Most of the research on weakly prime and weakly 2-absorbing modules is for modules over commutative rings. Only scatterd results about these notions with regard to non-commutative rings are available. The motivation of this paper is to show that many results for the commutative case also hold in the non-commutative case. Let \(R\) be a non-commutative ring with identity. We define the notions of a weakly prime and a weakly 2-absorbing submodules of \(R\) and show that in the case that \(R\) commutative, the definition of a weakly 2-absorbing submodule coincides with the original definition of A. Darani and F. Soheilnia. We give an example to show that in general these two notions are different. The notion of a weakly m-system is introduced and the weakly prime radical is characterized interms of weakly m-systems. Many properties of weakly prime submodules and weakly 2-absorbing submodules are proved which are similar to the results for commutative rings. Amongst these results we show that for a proper submodule \(N_i\) of an \(R_i\)-module \(M_i\), for \(i = 1 , 2\), if \(N_1 \times N_2\) is a weakly 2-absorbing submodule of \(M_1 \times M_2\), then \(N_i\) is a weakly 2-absorbing submodule of \(M_i\) for \(i = 1 , 2\).Quasi-dual Baer moduleshttps://www.zbmath.org/1483.160112022-05-16T20:40:13.078697Z"Tribak, Rachid"https://www.zbmath.org/authors/?q=ai:tribak.rachid"Talebi, Yahya"https://www.zbmath.org/authors/?q=ai:talebi.yahya"Hosseinpour, Mehrab"https://www.zbmath.org/authors/?q=ai:hosseinpour.mehrabSummary: Let \(R\) be a ring and let \(M\) be an \(R\)-module with \(S=\mathrm{End}_R(M)\). The module \(M\) is called quasi-dual Baer if for every fully invariant submodule \(N\) of \(M\), \(\{\phi\in S\mid \mathrm{Im}\phi\subseteq N\}= eS\) for some idempotent \(e\) in \(S\). We show that \(M\) is quasi-dual Baer if and only if \(\sum_{\varphi\in\mathfrak{I}}\varphi (M)\) is a direct summand of \(M\) for every left ideal \(\mathfrak{I}\) of \(S\). The \(R\)-module \(R_R\) is quasi-dual Baer if and only if \(R\) is a finite product of simple rings. Other characterizations of quasi-dual Baer modules are obtained. Examples which delineate the concepts and results are provided.On singular equivalences of Morita type with level and Gorenstein algebrashttps://www.zbmath.org/1483.160122022-05-16T20:40:13.078697Z"Dalezios, Georgios"https://www.zbmath.org/authors/?q=ai:dalezios.georgiosThe author studies singular equivalences of finite-dimensional algebras induced from tensor product functors, generalizing a result of Rickard over self-injective algebras. Specifically, necessary and sufficient conditions imposed over a complex bimodule X in order the functor \(X\otimes -\) to induce an equivalence of algebras.
As corollaries, he obtains a bimodule version of
\textit{S. Oppermann} et al. result [Adv. Math. 350, 190--241 (2019; Zbl 1470.16019)]; and over Gorenstein algebras, conditions to obtain a singular equivalence of Morita type with level.
Reviewer: Luz Adriana Mejia Castaño (Barranquilla)On \(F\)-\(Z\)-hollow and \(Z\)-semihollow moduleshttps://www.zbmath.org/1483.160132022-05-16T20:40:13.078697Z"Hamad, Amina T."https://www.zbmath.org/authors/?q=ai:hamad.amina-t"Elewi, Alaa A."https://www.zbmath.org/authors/?q=ai:elewi.alaa-aSummary: In this work, \(F\)-\(Z\)-hollow and \(Z\)-semihollow modules are defined as a generalization of \(F\)-hollow and semihollow modules, and some properties of these concepts of modules are investigated. Where an R-module \(E\) over a ring \(R\) is \(F\)-\(Z\)-hollow if every proper submodule of \(E\) is \(F\)-\(Z\)-small in \(E\). And we say that a nonzero \(R\)-module \(E\) is \(Z\)-semihollow module if every proper finitely generated submodule of \(E\) \(Z\)-small submodule.On the Hochschild homology of smash biproductshttps://www.zbmath.org/1483.160142022-05-16T20:40:13.078697Z"Kaygun, A."https://www.zbmath.org/authors/?q=ai:kaygun.atabey"Sütlü, S."https://www.zbmath.org/authors/?q=ai:sutlu.serkanSummary: We develop a new spectral sequence in order to calculate the Hochschild homology of smash biproducts (also called the twisted tensor products) of unital associative algebras \(A \# B\) provided one of \(A\) or \(B\) has Hochschild dimension less than 2. We use this spectral sequence to calculate Hochschild homology of the algebra \(M_q(2)\) of quantum \(2 \times 2\)-matrices.Rings whose (proper) cyclic modules have cyclic automorphism-invariant hullshttps://www.zbmath.org/1483.160152022-05-16T20:40:13.078697Z"Koşan, M. Tamer"https://www.zbmath.org/authors/?q=ai:kosan.muhammet-tamer"Quynh, Truong Cong"https://www.zbmath.org/authors/?q=ai:quynh.truong-congA module \(M\) is called automorphism-invariant if \(M\) is invariant under any automorphism of the injective hull of \(M\). It is shown that for a ring \(R\), if every cyclic right \(R\)-module is automorphism-invariant, then \(R\) is stably-finite (i.e., \(\text{Mat}_n(R)\) is directly-finite for each positive integer \(n\)).
The minimal automorphism-invariant essential extension of a module \(M\) is called the automorphism-invariant hull of \(M\). Further, a ring \(R\) is called right \(a\)-hypercyclic if each cyclic right \(R\)-module has a cyclic automorphism-invariant hull. Then the authors show that if a ring \(R\) is right \(a\)-hypercyclic and \(R_R\) has Krull dimension (in the sense of Rentschler and Gabriel), then \(R\) is right Artinian.
Reviewer: Jae Keol Park (Pusan)Dualities from iterated tiltinghttps://www.zbmath.org/1483.160162022-05-16T20:40:13.078697Z"Huisgen-Zimmermann, Birge"https://www.zbmath.org/authors/?q=ai:huisgen-zimmermann.birgeThis paper follows two goals. First it studies dualities for tilting modules in the full subcategory \(\mathcal{P}^{<\infty}(\Lambda\text{-mod})\) of modules of finite projective dimension over a finite-dimensional algebra \(\Lambda\). For \(T\), a tilting module of \(\Lambda\) and \(\widetilde{\Lambda}\), its corresponding tilted algebra, there is a duality coming from the work of \textit{Y. Miyashita} [Math. Z. 193, 113--146 (1986; Zbl 0578.16015)] between the two contravariant functors \[ \operatorname{Hom}_{\Lambda}(-,T) : \mathcal{P}^{<\infty}(\Lambda\text{-mod}) \longleftrightarrow \mathcal{P}^{<\infty}(\text{mod-}\widetilde{\Lambda}) : \operatorname{Hom}_{\widetilde{\Lambda}}(-,T) \] when \( _\Lambda T_{\widetilde{\Lambda}}\) is a tilting bimodule strong on both side, so if it is Ext-injective relative to objects of \(\mathcal{P}^{<\infty}(\Lambda\text{-mod})\) and \(\mathcal{P}^{<\infty}(\text{mod-}\widetilde{\Lambda})\). Theorem 1 of the article proves a converse statement in a more general context whose corollary is that, if there is such a pair of dualities between two finite-dimensional algebras, then they can be induced by a tilting bimodule strong on both sides.
The article then specialises its investigation to truncated path algebras. On this it follows a second goal: to exhibit the special behavior of these algebra in relation to tilting and compares them with hereditary algebras at the homological level. Notably it provides statements on projective dimension of modules in the bigger category \(\text{Mod-}\widetilde{\Lambda}\) (Theorem 12), on the passage of contravariant finiteness from \(\mathcal{P}^{<\infty}(\Lambda\text{-mod})\) to \(\mathcal{P}^{<\infty}(\text{mod-}\widetilde{\Lambda})\) (Theorem 17) and on the 2-periodicity of iterated strong tilting (Theorem 19).
This study builds upon previous works [\textit{A. Dugas} and the author, Manuscr. Math. 134, No. 1--2, 225--257 (2011; Zbl 1231.16013); \textit{K. Goodearl} and the author, Algebra Number Theory 12, No. 2, 379--410 (2018; Zbl 1420.16008)] and its applications are illustrated in later work [the author and \textit{M. Saorín}, Contemp. Math. 761, 61--101 (2021; Zbl 1470.16025)]. A nice feature of the article is example (3B) used through the text to ground the results.
Reviewer: Alexis Langlois-Rémillard (Gent)On \(\tau\)-tilting finiteness of the Schur algebrahttps://www.zbmath.org/1483.160172022-05-16T20:40:13.078697Z"Wang, Qi"https://www.zbmath.org/authors/?q=ai:wang.qi.4|wang.qi|wang.qi.5|wang.qi.1|wang.qi.6|wang.qi.2|wang.qi.3\textit{T. Adachi} et al. [Compos. Math. 150, No. 3, 415--452 (2014; Zbl 1330.16004)] introduced \(\tau\)-tilting theory. \textit{L. Demonet} et al. [Int. Math. Res. Not. 2019, No. 3, 852--892 (2019; Zbl 07130859)] initiated the study of \(\tau\)-tilting finite algebras. So far \(\tau\)-tilting finite algebras have been classified for many classes of finite dimensional algebras such as algebras with radical square zero, preprojective algebras of Dynkin type, Brauer graph algebras, biserial algebras, and minimal wild two-point algebras. In the paper under review, the author tries to classify \(\tau\)-tilting finite Schur algebras over an algebraically closed field of characteristic \(p>0\). He proves that all tame Schur algebras are \(\tau\)-tilting finite. For the wild Schur algebras \(S(n,r)\) with \(p=2,n=2,r=8,17,19\) or \(p=2,n=3,r=4\) or \(p=2,n\ge 5,r=5\) or \(p\ge 5, n=2, p^2\le r\le p^2+p-1\), he can not determine whether they are \(\tau\)-tilting finite or not at present since the number of pairwise non-isomorphic basic support \(\tau\)-tilting modules is huge for them, but he conjectures they are. Except for the cases above, he can show that a wild Schur algebra \(S(n,r)\) is \(\tau\)-tilting finite if and only if \(p=2,n=2,r=6,13,15\) or \(p=2,n=3,r=5\) or \(p=2,n=4,r=4\).
Reviewer: Yang Han (Beijing)Indecomposable non uniserial modules of length threehttps://www.zbmath.org/1483.160182022-05-16T20:40:13.078697Z"D'Este, Gabriella"https://www.zbmath.org/authors/?q=ai:deste.gabriellaSummary: We investigate a particular class of indecomposable modules of length three, defined over a \(K\)-algebra, with a simple socle and two non isomorphic simple factor modules. These modules may have any projective dimension different from zero. On the other hand their composition factors may have any countable dimension as vector spaces over the underlying field \(K\). Moreover their endomorphism rings are \(K\)-vector spaces of dimension \(\leq 2\).Baer-Kaplansky theorem for modules over non-commutative algebrashttps://www.zbmath.org/1483.160192022-05-16T20:40:13.078697Z"D'Este, Gabriella"https://www.zbmath.org/authors/?q=ai:deste.gabriella"Tütüncü, Derya Keskin"https://www.zbmath.org/authors/?q=ai:keskin-tutuncu.deryaSummary: In this paper we investigate the Baer-Kaplansky theorem for module classes on algebras of finite representation types over a field. To do this we construct finite dimensional quiver algebras over any field.Lecture notes on quivers with superpotential and their representationshttps://www.zbmath.org/1483.160202022-05-16T20:40:13.078697Z"Quintero Vélez, Alexander"https://www.zbmath.org/authors/?q=ai:quintero-velez.alexander"Valencia, Fabricio"https://www.zbmath.org/authors/?q=ai:valencia.fabricioSummary: These lecture notes are based on a mini-course presented at the fifth version of the Workshop Geometry in Algebra and Algebra in Geometry held in Medellín-Colombia in October 2019. The aim is to provide the background necessary to understand the theory of quivers with relations given by superpotentials. A heavy emphasis is placed throughout on examples to illustrate the applicability of the theory. The motivations for the lectures come from several sources: superpotentials in physics, Calabi-Yau algebras, and noncommutative resolutions.Schofield sequences in the Euclidean casehttps://www.zbmath.org/1483.160212022-05-16T20:40:13.078697Z"Szántó, Csaba"https://www.zbmath.org/authors/?q=ai:szanto.csaba"Szöllősi, István"https://www.zbmath.org/authors/?q=ai:szollosi.istvanSummary: Let \(k\) be a field and consider the path algebra \textit{kQ} of the quiver \(Q\). A pair of indecomposable \textit{kQ}-modules \((Y, X)\) is called an orthogonal exceptional pair if the modules are exceptional and \(\mathrm{Hom}(X,Y)=\mathrm{Hom}(Y, X)=\mathrm{Ext}^1(X,Y)=0\). Denote by \(\mathcal{F}(X, Y)\) the full subcategory of objects having a filtration with factors \(X\) and \(Y\). By a theorem of Schofield if \(Z\) is exceptional but not simple, then \(Z \in \mathcal{F}(X, Y)\) for some orthogonal exceptional pair \((Y, X)\), and \(Z\) is not a simple object in \(\mathcal{F}(X, Y)\). In fact, there are precisely \(s(Z) - 1\) such pairs, where \(s(Z)\) is the support of \(Z\) (i.e. the number of nonzero components in \(\dim_{\_} Z)\). Whereas it is easy to construct \(Z\) given \(X\) and \(Y\), there is no convenient procedure yet to determine the possible modules \(X\) (called Schofield submodules of \(Z)\) and then \(Y\) (called Schofield factors of \(Z)\), when \(Z\) is given. We present such an explicit procedure in the tame case, i.e. when \(Q\) is Euclidean.On Tate duality and a projective scalar property for symmetric algebrashttps://www.zbmath.org/1483.160222022-05-16T20:40:13.078697Z"Eisele, Florian"https://www.zbmath.org/authors/?q=ai:eisele.florian"Geline, Michael"https://www.zbmath.org/authors/?q=ai:geline.michael"Kessar, Radha"https://www.zbmath.org/authors/?q=ai:kessar.radha"Linckelmann, Markus"https://www.zbmath.org/authors/?q=ai:linckelmann.markusSummary: We identify a class of symmetric algebras over a complete discrete valuation ring \(\mathcal{O}\) of characteristic zero to which the characterisation of Knörr lattices in terms of stable endomorphism rings in the case of finite group algebras can be extended. This class includes finite group algebras, their blocks and source algebras and Hopf orders. We also show that certain arithmetic properties of finite group representations extend to this class of algebras. Our results are based on an explicit description of Tate duality for lattices over symmetric \(\mathcal{O}\)-algebras whose extension to the quotient field of \(\mathcal{O}\) is separable.Images of multilinear polynomials in the algebra of finitary matrices contain trace zero matriceshttps://www.zbmath.org/1483.160232022-05-16T20:40:13.078697Z"Vitas, Daniel"https://www.zbmath.org/authors/?q=ai:vitas.danielLet \(F\) be an infinite field and let \(F\langle X_1, \dots, X_n\rangle\) denote the free associative algebra, freely generated by the set \(X=\{ X_1, \dots, X_n\}\). The elements of \(F\langle X_1, \dots, X_n \rangle\) are called noncommutative polynomials. A polynomial \(f=f(x_1, \dots, x_n)\) is called multilinear if \(f\) can be written as
\[
f(x_1, \dots, x_n)=\sum_{\sigma\in S_n}\lambda_\sigma X_{\sigma(1)}\cdots X_{\sigma(n)},
\]
where \(S_n\) denotes the symmetric group and \(\lambda_\sigma\in F\).
For a given \(F\)-algebra \(A\), a polynomial \(f(x_1,\dots,x_n)\in F\langle X_1, \dots, X_n\rangle\) defines a map (also denoted by \(f\))
\[
\begin{array}{cccc} f: & A^n & \longrightarrow & A \\
& (a_1,\dots,a_n) & \mapsto & f(a_1,\dots,a_n) \\
\end{array}
\]
The image of such map \(f\) is called the image of the polynomial \(f\) on \(A\).
The most important and long-standing problem related to images of polynomials is the so called Lvov-Kaplansky conjecture. It states that the image of a multilinear polynomial \(f\) on \(M_d(F)\) is a vector space, which is equivalent to the image of \(f\) being one of the following: \(\{0\}\), \(F\) (viewed as the set of scalar matrices), \(sl_d(F)\) (the set of trace zero matrices) or \(M_d(F)\).
Solutions to such problem are known only for \(d=2\) [\textit{A. Kanel-Belov} et al., Proc. Am. Math. Soc. 140, No. 2, 465--478 (2012; Zbl 1241.16017)] if \(F\) is quadratically closed. The case \(d=3\) has interesting advances, but not a solution up to now.
The infinite-dimensional analogue of such conjecture (i.e., for \(A=\mathrm{End}(V)\), where \(V\) is an infinite-dimensional vector space over \(F\)) was proved by the author in [\textit{D. Vitas}, J. Algebra 565, 255--281 (2021; Zbl 1459.16027)], and states that an arbitrary nonzero multilinear polynomial is surjective on \(A\).
In the paper under review, the author studies the similar question for the algebra \(M_\infty (F)\), of finitary matrices over \(F\), i.e., countably infinite matrices with only finitely many nonzero entries.
The main theorem of the paper implies that if \(F\) is an infinite field and \(d\in \mathbb{N}\), then for every nonzero multilinear polynomial \(f\), there exists an \(s\in\mathbb{N}\) such that the image of \(f\) on \(M_s(F)\) contains \(sl_d(F)\).
The above inclusion should be understand as \[M_d(F)=\begin{pmatrix} M_d(F) & 0 \\
0 & 0 \end{pmatrix}\subseteq M_s(F).\]
This result is similar to the statement of the so called Mesyan Conjecture [\textit{Z. Mesyan}, Linear Multilinear Algebra 61, No. 11, 1487--1495 (2013; Zbl 1290.15011)], a weaker form of the Lvov-Kaplansky Conjecture, which states that if \(d\geq n-1\) the image of a multilinear polynomial of degree \(n\) on \(M_d(F)\) contains \(sl_d(F)\). Such conjecture was proved for \(d=3\) by Mesyan and for \(d=4\) by Buzinsky and Winstanley. It is worth mentioning that the case \(d=4\) contains an error in its proof, but the result is still valid.
An immediate consequence of the above theorem is that the analog of the Mesyan conjecture for \(M_\infty(F)\) is true. Namely, if \(F\) is an infinite field and \(f\) is a nonzero multilinear polynomial, then the image of \(f\) on \(M_\infty(F)\) contains \(sl_\infty(F)\) (the set of trace zero finitary matrices). Also, the reverse inclusion holds if and only if \(f\) is a sum of commutators.
The paper uses similar methods and ideas as the ones in [\textit{D. Vitas}, J. Algebra 565, 255--281 (2021; Zbl 1459.16027)].
Reviewer: Thiago Castilho de Mello (São José dos Campos)Stability in graded rings associated with commutative augmented ringshttps://www.zbmath.org/1483.160242022-05-16T20:40:13.078697Z"Chang, Shan"https://www.zbmath.org/authors/?q=ai:chang.shanA commutative augmented ring is defined to be a commutative ring \(A\) together with a homomorphism \( \varepsilon : A \rightarrow \mathbb {Z }\) which satisfies:
\begin{itemize}
\item A has an identity element and \( \varepsilon\) preserves identity elements,
\item The additive group of the ring \(A\) is finitely generated free abelian,
\item \(I/I^2\) is torsion as an additive group, where \(I = ker \, \varepsilon \).
\end{itemize}
The homomorphism \( \varepsilon \) is called the augmentation map of \(A\) and the kernel \(I\) of \( \varepsilon\) is called the augmentation ideal of \(A\). If \(G\) is a finite abelian group then the integral group ring \( \mathbb {Z } G\) is a classical commutative augmented ring, where its augmentation map is induced by sending \(g\) to \(1\) for each element \(g \in G\).
In the paper, \(Q_n(A)\) denotes the \(n\)-th consecutive quotient group \(I^n/I^{n+1}\), where \(I^n\) is the \(n\)-th power of \(I\). The main theorem of the paper states that if \(A\) is a commutative augmented ring then there exists a positive integer \(n_0\) such that \(Q_n(A) \cong Q_{n+1}(A)\) for any \(n \geq n_0\).
Reviewer: Anatolii Tushev (Dnipro)Unit groups of group algebras of groups of order 18https://www.zbmath.org/1483.160252022-05-16T20:40:13.078697Z"Sahai, Meena"https://www.zbmath.org/authors/?q=ai:sahai.meena"Ansari, Sheere Farhat"https://www.zbmath.org/authors/?q=ai:ansari.sheere-farhatThe study of the structure of unit groups of a group algebra is an important and interesting study within the larger area of study called group algebras.
Here, the authors study the structure of unit groups of group algebras of groups of order 18 over a finite field \(F\) of order \(q=p^{n}\), where the prime \(p\), is the characteristic of \(F\).
The main results are five lengthy structure theorems and their proofs for the following five unit groups \(U(FC_{18})\), \(U(FD_{18})\), \(U(F(D_{6} \times C_{3}))\), \(U(F(C_{6} \times C_{3}))\) and \(U(F(C^{2}_{3} \rtimes C_{2}))\) where \(D_{n}\) is the dihedral group of order \(n\) and \(C_{n}\) is the cyclic group of order \(n\).
The question arises here is that what would be the structure of the above unit groups for an integral group ring or some general group rings? The answer to this question would be much more interesting than that of a finite group algebra. Also, a good thing about this article is that there are not too many references.
Reviewer: Telveenus Antony (Kingston upon Thames)Maximal prime homomorphic images of mod-\(p\) Iwasawa algebrashttps://www.zbmath.org/1483.160262022-05-16T20:40:13.078697Z"Woods, William"https://www.zbmath.org/authors/?q=ai:woods.williamLet \(kG\) be the completed group ring of a compact \(p\)-adic analytic group \(G\) over a finite field \(k\) of characteristic \(p\) and let \(P\) be a minimal prime ideal of \(kG\). Denote a some finite field extension by \(k'/k\), the large subquotient of \(G\) with no finite normal subgroups by \(G'\) and and a ``twisting'' operation that preserves many desirable properties of the ring structure by \((-)_{\alpha}\). The author gives the structure of the ring \(kG/P\) and an explicit isomorphism between \(kG/P\) and a matrix ring with coefficients in the ring \((k'G')_{\alpha}\). He uses this isomorphism and studies the correspondence induced between certain ideals of \(kG\) and of \((k'G')_{\alpha}\) and shows, that this preserves many useful ``group-theoretic'' properties of ideals.
Reviewer: Todor Mollov (Plovdiv)A quaternionic Nullstellensatzhttps://www.zbmath.org/1483.160272022-05-16T20:40:13.078697Z"Alon, Gil"https://www.zbmath.org/authors/?q=ai:alon.gil"Paran, Elad"https://www.zbmath.org/authors/?q=ai:paran.eladSummary: We prove a Nullstellensatz for the ring of polynomial functions in \(n\) non-commuting variables over Hamilton's ring of real quaternions. We also characterize the generalized polynomial identities in \(n\) variables which hold over the quaternions, and more generally, over any division algebra.Leavitt path algebras of weighted Cayley graphs \(C_n(S,w)\)https://www.zbmath.org/1483.160282022-05-16T20:40:13.078697Z"Mohan, R."https://www.zbmath.org/authors/?q=ai:mohan.ravi|mohan.ram-v|mohan.raghuveer|mohan.ranjith|mohan.ranju|mohan.radha|mohan.radhe|mohan.rakhi|mohan.rakesh|mohan.rohitConsider a finite group \(G\), a subset \(S\subset G\) with \(S\) any nonempty generating set of \(G\), a map \(w\colon S \to {\mathbb N}\) and \(\hbox{Cay}(G,S,w)\) the weighted Cayley graph. In the particular case that \(n\) is a positive integer and \(G={\mathbb Z}_n\) one can denote \(C_n(S,w)=\hbox{Cay}({\mathbb Z}_n,S,w)\). In Theorem 3.1 of this paper, the author finds that \(L(\hbox{Cay}(G,S,w))\) is a purely infinite simple Leavitt path algebra if and only if \(W:=\sum_{s \in S}w(s) \ge 2\) and if and only if \(L(\hbox{Cay}(G,S,w))\) does not have the IBN property. In the case \(S=\{s_1,s_2, \ldots , s_k\} \subsetneq {\mathbb Z}_n, \ s_1<s_2< \cdots <s_k\), \(0 \notin S\) and \(\sum_{s_j \in S}w(s_j) \ge 2\) (\(L(C_n(S,w))\) a purely infinite simple Leavitt path algebra), the author shows how to calculate its Grothendieck group. Also other specific Leavitt path algebras of Cayley graphs and its Grothendieck group \(K_0(L(C_n(S,w)))\) are described. So is the case of Leavitt path algebras associated to Cayley graphs for dihedral groups.
Reviewer: Dolores Martín Barquero (Málaga)A classification of ideals in Steinberg and Leavitt path algebras over arbitrary ringshttps://www.zbmath.org/1483.160292022-05-16T20:40:13.078697Z"Rigby, Simon W."https://www.zbmath.org/authors/?q=ai:rigby.simon-w"van den Hove, Thibaud"https://www.zbmath.org/authors/?q=ai:van-den-hove.thibaudThe authors give a one-to-one correspondence between ideals in the Steinberg algebra of a Hausdorff ample groupoid \(G\), and certain families of ideals in the group algebras of isotropy groups in \(G\). In any small category \(C\), taking any \(u\in C\) (an object), the set \(\hom_C(u,u)\) is a monoid (relative to composition) and for a groupoid \(G\) it is a group denoted \(G_u^u\). Then it has sense to consider the group algebra \(RG_u^u\). If \(A_R(G)\) is the Steinberg \(R\)-algebra and \(u\in G^{(0)}\), \(f\in A_R(G)\) the authors define in the paper \(f_u:=f\vert_{G_u^u}\in RG_u^u\). Next, it is considered in the paper the set \(X(G)=\cup_{u\in G^{(0)}}RG_u^u\) which is topologized suitably (Lemma 3.1). Then the so called disassembly map: \[I \mapsto Y_I := \cup_{u\in G^{(0)}}I_u, \hbox{ where } I_u := \{f_u\colon f \in I\}\subset RG_u^u. \] The main theorem of the paper proves that this map defines a bijection between ideals in the Steinberg algebra of a Hausdorff ample groupoid \(G\), and a certain set of all open subsets \(Y\subset X(G)\) satisfying specified conditions (Theorem 3.2). Then, this is used to give a complete graph-theoretic description of the ideal lattice of Leavitt path algebras over arbitrary commutative rings, generalising the classification of ideals in Leavitt path algebras over fields.
Reviewer: Candido Martín González (Málaga)On the tensor product of \(m\)-partition algebrashttps://www.zbmath.org/1483.160302022-05-16T20:40:13.078697Z"Kennedy, A. Joseph"https://www.zbmath.org/authors/?q=ai:kennedy.a-joseph"Jaish, P."https://www.zbmath.org/authors/?q=ai:jaish.pSummary: We study the tensor product algebra \(P_k (x_1) \otimes P_k (x_2) \otimes \cdots\otimes P_k (x_m)\), where \(P_k (x)\) is the partition algebra defined by Jones and Martin. We discuss the centralizer of this algebra and corresponding Schur-Weyl dualities and also index the inequivalent irreducible representations of the algebra \(P_k (x_1) \otimes P_k (x_2) \otimes \cdot \cdot \cdot \otimes P_k (x_m)\) and compute their dimensions in the semisimple case. In addition, we describe the Bratteli diagrams and branching rules. Along with that, we have also constructed the RS correspondence for the tensor product of \(m\)-partition algebras which gives the bijection between the set of tensor product of \(m\)-partition diagram of \(P_k (n_1) \otimes P_k (n_2) \otimes \cdots\otimes P_k (n_m)\) and the pairs of \(m\)-vacillating tableaux of shape \([\lambda] \in \Gamma_k^m, \Gamma_k^m = \{[\lambda] = (\lambda_1, \lambda_2,\dots, \lambda_m)\mid \lambda_i \in \Gamma_k, i \in \{1, 2,\dots, m \} \}\) where \(\Gamma_k = \{\lambda_i \vdash t\mid 0 \leq t \leq k \}\). Also, we provide proof of the identity \((n_1 n_2 \cdot \cdot \cdot n_m)^k = \sum_{[\lambda] \in \Lambda_{n_1, n_2, \dots, n_m}^k} f^{[\lambda]} m_k^{[\lambda]}\) where \(m_k^{[\lambda]}\) is the multiplicity of the irreducible representation of \(S_{n_1} \times S_{n_2} \times \dots \times S_{n_m}\) module indexed by \([\lambda] \in \Lambda_{n_1, n_2, \dots, n_m}^k\), where \(f^{[\lambda]}\) is the degree of the corresponding representation indexed by \([\lambda] \in \Lambda_{n_1, n_2, \dots, n_m}^k\) and \([\lambda] \in \Lambda_{n_1, n_2, \dots, n_m}^k = \{[\lambda] = (\lambda_1, \lambda_2, \dots, \lambda_m)\mid \lambda_i \in \Lambda_{n_i}^k, i \in \{1, 2, \dots, m \} \}\) where \(\Lambda_{n_i}^k = \{\mu = (\mu_1, \mu_2, \dots, \mu_t) \vdash n_i\mid n_i - \mu_1 \leq k \}\).Galois and cleft monoidal cowreaths. Applicationshttps://www.zbmath.org/1483.160312022-05-16T20:40:13.078697Z"Bulacu, D."https://www.zbmath.org/authors/?q=ai:bulacu.daniel"Torrecillas, B."https://www.zbmath.org/authors/?q=ai:torrecillas.blasThe theory of Hopf-Galois extensions and weak cleft extensions has been developed in the last decades for various Hopf algebraic structures such as Hopf algebras, weak Hopf algebras, Hopf algebroids, Frobenius Hopf algebroids, Hopf quasigroups, weak Hopf quasigroups, etc. Also, the study of this theory was performed in a general monoidal setting and in the literature we can find extensions of the Hopf-Galois theory to entwining structures and weak entwining structures. In all the previous cases the main objective has been to characterize cleft extensions as Galois extensions with the normal basis property (this property is a generalization of the normal basis property of an extension of fields) or as some kind of cross product defined by an action and a cocycle. Taking all this into account, the main motivation of this book is to show how cowreaths in monoidal categories are the tool that permits to unify a considerable part of the above theories and results and on the other hand to introduce a theory of Hopf-Galois and cleft extensions for quasi-Hopf algebras since in this context such theory did not exist.
Let \({\mathcal C}\) be a monoidal category. A cowreath is a comonad in the Eilenberg-Moore category associated to \({\mathcal C}\). If we denote it by \(EM({\mathcal C})\), a cowreath is a couple \((A,X)\) where \(A\) is an algebra in \({\mathcal C}\) and \(X\) is a coalgebra in the monoidal category \({\mathcal T}^{\sharp}_{A}:=EM({\mathcal C})(A)\). As was pointed by the authors, any cowreath \((A,X)\) admits a category of entwined modules, denoted by \({\mathcal C}(\psi)_{A}^{X}\), and \((A,X)\) is called pre-Galois if \(A\in {\mathcal C}(\psi)_{A}^{X}\). Therefore, \((A,X)\) is pre-Galois if the coalgebra part in \({\mathcal T}^{\sharp}_{A}\) admits an almost group-like element. The existence of this element permits to define the subalgebra of coinvariants of \(A\), \(B=A^{co(X)}\), as well as the canonical morphisms \(\mathrm{can}\;:\;A\otimes_{B}A\rightarrow A\otimes X\) under suitable conditions, for example if the category \({\mathcal C}\) admits equalizers and any object is coflat and robust. The canonical morphisms can be defined for any object \({\mathfrak M}\) in the category \({\mathcal T}^{\sharp}_{A}\) and, as was proved by the authors, can is an isomorphism if, and only if, it is an isomorphism for any object in \({\mathcal T}^{\sharp}_{A}\). When the latter happens it will be said, by analogy with the Hopf algebra case, that \((A,X)\) is a Galois cowreath. Under these conditions it is possible to guarantee that the functor \(L:=-\otimes_{B}A:{\mathcal C}_{B}\rightarrow {\mathcal C}(\psi)_{A}^{X}\) admits as a right adjoint the functor of coinvariants \(R=(-)^{co(X)}\) (see Theorem 3.10). Then, as a consequence, the authors prove in Theorem 4.9 that the adjuntion \(L\dashv R\) is an equivalence of categories if, and only if, \((A,X)\) is a Galois cowreath and the functor \(-\otimes_{B}A:{\mathcal C}_{B}\rightarrow {\mathcal C}\) preserves and reflects equalizers, i.e., the object \(_{A}B\) is faithfully flat. Next, in Chapter 5, the authors find sufficient conditions to obtain that \(L\dashv R\) is an equivalence. More concretely, in Theorem 5.10 they prove that any Galois cowreath \((A,X)\), satisfying that the coalgebra part is coseparable in the category \({\mathcal T}^{\sharp}_{A}\), gives rise to the desired categorical equivalence. Also, they obtain that the coseparability is linked with the existence of a total integral \(\lambda:X\rightarrow A\) and, as a consequence, any object of \({\mathcal C}(\psi)_{A}^{X}\) is \(A\)-relative injective and, moreover, \(A\in\) \(_{B}{\mathcal C}(\psi)_{A}^{X}\) is \(_{B}{\mathcal C}_{A}\)-relative injective. Therefore, under these new conditions, the existence of a total integral or the conditions of relative injectivity, \(L\dashv R\) is an equivalence of categories.
As was pointed in the first paragraph of this review, it is a well-known fact (proved by \textit{Y. Doi} and \textit{M. Takeuchi} [Commun. Algebra 14, 801--817 (1986; Zbl 0589.16011)]) that in the classical Hopf-Galois theory Galois extensions with the normal basis property are nothing but cleft extensions. The main target of the sixth chapter of the book is to prove a similar result for cowreaths. To achieve it the authors introduce the notion of cleft cowreath requiring the existence of two morphisms \(\Phi\), \(\phi\) \(:X\rightarrow A\) satisfying the conditions contained in Definition 6.1. This is an interesting definition because if \(X\) is a coalgebra in \({\mathcal T}_{A}\) the conditions of the previous definition reduces to the clasical conditions for the so called cleaving morphism. On the other hand, it is applicable to the quasi-Hopf algebra setting since quasi-Hopf algebras give rise to examples of cowreaths in \({\mathcal T}^{\sharp}_{A}\) and not in \({\mathcal T}_{A}\). Finally, in Theorem 6.8 the authors prove the following: Let \((A,X)\) be a pre-Galois cowreath satisfying the condition 6.3 (see page 57) and such that \(X\) is a left flat object in the category \({\mathcal C}\). Then, \((A,X)\) is cleft iff it is Galois and satisfies the normal basis property.
In Chapter 7, the authors present a one to one correspondence between cleft cowreaths and cleft wreaths in a monoidal category. In the Hopf algebra context cleft extensions can be identified to a sort of crossed products defined by an action and a convolution invertible cocycle. In this chapter they show that for cowreaths in place of the classical crossed products, it is more convenient to consider wreath algebras in the sense of \textit{S. Lack} and \textit{R. Street} [J. Pure Appl. Algebra 175, No. 1--3, 243--265 (2002; Zbl 1019.18002)]), equipped with two additional morphisms satisfying the conditions contained in (7.6). The authors call these wreaths cleft and in Theorem 7.11 and Corollary 7.12 they prove that these kind of wreaths describe completely the cleft cowreaths, up to a unitally cowreath isomorphism.
The final part of the book is dedicated to the applications of the results contained in the previous chapters to Galois-cleft theory for quasi-Hopf algebras. For example in Chapter 8 for a quasibialgebra \(H\), a right \(H\)-comodule algebra \({\mathfrak A}\) and a right \(H\)-module coalgebra \(C\) the authors explain the structure of the cowreath \(({\mathfrak A}, C)\) and when it is Galois. Also, they find the conditions under which we can obtain the categorical equivalence presented in Theorem 4.9. Moreover, in Chapter 12, the authors prove that right comodule algebras over \(H\) induces examples of Galois cowreaths satisfying the normal basis property and therefore they can guarantee that in this context there exist examples of cleft cowreaths.
In Chapter 9, the authors investigate when a pre-Galois cowreath defined by an entwining structure \((A,C)_{\psi}\) is cleft. They prove in Theorem 9.6 that this happens iff the algebra \(A\) is isomorphic as algebra and as right \(C\)-comodule to a crossed product by a coalgebra \({\mathfrak B}\sharp C\) for which the canonical embedding morphism from \(C\) to \({\mathfrak B}\sharp C\) is convolution invertible. Chapter 10 and 11 are dedicated to the study of cowreaths associated to \(\nu\)-Doi-Hopf modules and to the study of those who appear when we work with generalized crossed products.
Finally, as the authors state, the results presented in this book can recover somehow of the Hopf-Galois and cleft theory developed by \textit{G. Böhm} and \textit{T. Brzeziński} [Appl. Categ. Struct. 14, No. 5--6, 431--469 (2006; Zbl 1133.16024)] for Hopf algebroids, the Schneider type theorem (see [\textit{A. Ardizzoni} et al., J. Algebra 321, No. 6, 1786--1796 (2009; Zbl 1165.16305)]), and consequently the Hopf-Galois theory for weak Hopf algebras. Also, it can be applied to braided Hopf algebras, Hom-Hopf algebras [\textit{S. Caenepeel} and \textit{I. Goyvaerts}, Commun. Algebra 39, No. 6, 2216--2240 (2011; Zbl 1255.16032)] and to Hopf group (co)algebras [\textit{S. Caenepeel} and \textit{M. De Lombaerde}, Commun. Algebra 34, No. 7, 2631--2657 (2006; Zbl 1103.16024)].
Reviewer: Ramón González Rodríguez (Vigo)A five-term exact sequence for Kac cohomologyhttps://www.zbmath.org/1483.160322022-05-16T20:40:13.078697Z"Galindo, César"https://www.zbmath.org/authors/?q=ai:galindo.cesar"Morales, Yiby"https://www.zbmath.org/authors/?q=ai:morales.yibySummary: We use relative group cohomologies to compute the Kac cohomology of matched pairs of finite groups. This cohomology naturally appears in the theory of abelian extensions of finite dimensional Hopf algebras. We prove that Kac cohomology can be computed using relative cohomology and relatively projective resolutions. This allows us to use other resolutions, besides the bar resolution, for computations. We compute, in terms of relative cohomology, the first two pages of a spectral sequence which converges to the Kac cohomology and its associated five-term exact sequence. Through several examples, we show the usefulness of the five-term exact sequence in computing groups of abelian extensions.Hopf algebra gauge theory on a ribbon graphhttps://www.zbmath.org/1483.160332022-05-16T20:40:13.078697Z"Meusburger, Catherine"https://www.zbmath.org/authors/?q=ai:meusburger.catherine"Wise, Derek K."https://www.zbmath.org/authors/?q=ai:wise.derek-kThere exist a variety of gauge theory-like models (for example models obtained from canonical quantization of Chern-Simons theory, models in 3d quantum gravity) constructed from algebraic data assigned to discretizations of oriented surfaces, many of which are based on Hopf algebras. These models strongly resemble lattice gauge theories, exhibiting gauge-like symmetries,
In the paper under review the authors give a ``a physically-motivated general definition of local Hopf algebra gauge theory on a ribbon graph, construct such gauge theories, and relate them to established models arising from quantizing Chern-Simons theory.''
Reviewer: Dmitry Artamonov (Moskva)Flatness of Noetherian Hopf algebras over coideal subalgebrashttps://www.zbmath.org/1483.160342022-05-16T20:40:13.078697Z"Skryabin, Serge"https://www.zbmath.org/authors/?q=ai:skryabin.sergeAn algebra over a fixed field \(k\) is said to be residually finite dimensional if its ideals of finite codimension have zero intersection. Let \(H\) be a Hopf algebra. In the paper under review, the author proves that, if \(A\) is a right Noetherian \(H\)-semiprime \(H\)-module algebra such that the action of \(H\) on \(A\) is locally finite, the algebra \(A\) has a right Artinian classical right quotient ring. Then, if \(A\) is a right Noetherian right coideal subalgebra of a residually finite dimensional Noetherian Hopf algebra \(H\), the algebra \(A\) has a right Artinian classical right quotient ring, and \(H\) is left \(A\)-flat.
As a consequence of the previous results it is also proved that the antipode of either right or left Noetherian residually finite-dimensional Hopf algebra is bijective. Hence, \(H\) is right and left Noetherian simultaneously.
Reviewer: Ramón González Rodríguez (Vigo)Generalised quantum determinantal rings are maximal ordershttps://www.zbmath.org/1483.160352022-05-16T20:40:13.078697Z"Lenagan, T. H."https://www.zbmath.org/authors/?q=ai:lenagan.thomas-h"Rigal, L."https://www.zbmath.org/authors/?q=ai:rigal.laurentSummary: Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.Skew braces as remnants of co-quasitriangular Hopf algebras in suplathttps://www.zbmath.org/1483.160362022-05-16T20:40:13.078697Z"Ghobadi, Aryan"https://www.zbmath.org/authors/?q=ai:ghobadi.aryanSkew braces are sets with two compatible group structures and they have been introduced by \textit{L. Guarnieri} and \textit{L. Vendramin} [Math. Comput. 86, No. 307, 2519--2534 (2017; Zbl 1371.16037)] to study set-theoretical solutions of the Yang-Baxter equation.
Considering linear solutions on vector spaces, there exists a relation between these kind of solutions and (co-)quasitriangular Hopf algebras. Into the specific, (co-)quasitriangular Hopf algebras and bialgebras provide solutions via their (co-)representation theory. Conversely, the Fadeev-Reshetikhin-Takhtajan construction produces such a bialgebra from any linear solution [\textit{N. Yu. Reshetikhin} et al., Leningr. Math. J. 1, No. 1, 193--225 (1990; Zbl 0715.17015); translation from Algebra Anal. 1, No. 1, 178--206 (1989)]. A question is whether skew braces can be viewed as Hopf algebras in a suitable category related to sets.
In the paper under review, the author finds the correct category to consider, that is the category SupLat of complete lattices and join preserving morphisms. Specifically, it is shown that any Hopf algebra \(H\) in SupLat, has a corresponding group \(R(H)\), and a co-quasitriangular structure on \(H\) induces a solution on \(R(H)\), which is compatible with its group structure. Conversely, any group with a compatible solution can be realised in this way.
Reviewer: Marzia Mazzotta (Lecce)Erratum to: ``On skew braces and their ideals''https://www.zbmath.org/1483.160372022-05-16T20:40:13.078697Z"Konovalov, A."https://www.zbmath.org/authors/?q=ai:konovalov.olexandr"Smoktunowicz, A."https://www.zbmath.org/authors/?q=ai:smoktunowicz.agata"Vendramin, L."https://www.zbmath.org/authors/?q=ai:vendramin.leandroErratum to the authors' paper [ibid. 30, No. 1, 95--104 (2021; Zbl 1476.16036)].A property satisfying reducedness over centershttps://www.zbmath.org/1483.160382022-05-16T20:40:13.078697Z"Jin, Hailan"https://www.zbmath.org/authors/?q=ai:jin.hailan"Kwak, Tai Keun"https://www.zbmath.org/authors/?q=ai:kwak.tai-keun"Lee, Yang"https://www.zbmath.org/authors/?q=ai:lee.yang"Piao, Zhelin"https://www.zbmath.org/authors/?q=ai:piao.zhelinIn this manuscript, the authors studied a ring property called pseudo-reduced-on-center that is satisfied by free algebras on commutative reduced rings. The authors have mainly investigated the structure of pseudo-reduced-over-center rings in relation to centers, radicals, and closely related concepts. Further, they also demonstrated that for pseudo-reduced-over-center rings of nonzero characteristic, the centers, and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. Conclusively it was drawn that if \(R\) is pseudo-reduced-over-center, then \(R\) is commutative and \(R/J(R)\) is a commutative regular ring with \(J(R)\) nil, where \(J(R)\) is the Jacobson radical of \(R\) and \(R\) a locally finite ring.
Reviewer: Mohd Arif Raza (Rabigh)A note on nil-clean ringshttps://www.zbmath.org/1483.160392022-05-16T20:40:13.078697Z"Danchev, Peter V."https://www.zbmath.org/authors/?q=ai:danchev.peter-vassilevTwo idempotents \(e\), \(f\) in a ring \(R\) generate the same left ideal (i.e., \( Re=Rf\)) iff \(ef=e\) and \(fe=f\). Such idempotents are called by the author, ``left-right symmetric''.
For any nil-clean ring it is well known that \(2\) is nilpotent, the Jacobson radical is nil and if the ring is also an abelian ring, every nilpotent element is contained in the Jacobson radical (see [\textit{A. J. Diesl}, J. Algebra 383, 197--211 (2013; Zbl 1296.16016)]). Repeatedly using these facts, the author obtains the following results:
Proposition 1. If \(R\) is a nil-clean ring such that each nilpotent is a difference of two commuting idempotents, then \(R\) is a boolean ring.
Theorem 1. Every nil-clean ring in which all nilpotents are difference of two left-right symmetric idempotents are strongly \(\pi \)-regular.
Using results from [the author and \textit{T.-Y. Lam}, Publ. Math. 88, No. 3--4, 449--466 (2016; Zbl 1374.16089)], the following result is derived
Theorem 2. Suppose \(R\) is a nil-clean ring with cyclic unit group \( U(R)\). Then \(R\) is strongly nil-clean of characteristic 2 if and only if \(U(R)\) is a 2-group.
In closing, some open problems are stated.
Reviewer: Grigore Călugăreanu (Cluj-Napoca)Compressed zero-divisor graphs of matrix rings over finite fieldshttps://www.zbmath.org/1483.160402022-05-16T20:40:13.078697Z"Đurić, Alen"https://www.zbmath.org/authors/?q=ai:duric.alen"Jevđenić, Sara"https://www.zbmath.org/authors/?q=ai:jevdenic.sara"Stopar, Nik"https://www.zbmath.org/authors/?q=ai:stopar.nikSummary: We extend the notion of the compressed zero-divisor graph \(\varTheta (R)\) to noncommutative rings in a way that still induces a product preserving functor \(\varTheta\) from the category of finite unital rings to the category of directed graphs. For a finite field \(F\), we investigate the properties of \(\varTheta (M_n (F))\), the graph of the matrix ring over \(F\), and give a purely graph-theoretic characterization of this graph when \(n \neq 3\). For \(n \neq 2\) we prove that every graph automorphism of \(\varTheta (M_n (F))\) is induced by a ring automorphism of \(M_n (F)\). We also show that for finite unital rings \(R\) and \(S\), where \(S\) is semisimple and has no homomorphic image isomorphic to a field, if \(\varTheta (R) \cong \varTheta (S)\), then \(R \cong S\). In particular, this holds if \(S= M_n (F)\) with \(n \neq 1\).Bases for skew derivationshttps://www.zbmath.org/1483.160412022-05-16T20:40:13.078697Z"Chuang, Chen-Lian"https://www.zbmath.org/authors/?q=ai:chuang.chen-lianLet \(R\) be a prime ring, \(Q\) its Utumi quotient ring, \(X\) a set of non-commuting variables intended to range over \(Q\). Let \(\Omega=\bigcup_{n\geq 0}\Omega_n\), where \(\Omega_n\subseteq \Omega_{n+1}\) for each \(n\geq 0\), be an expansion closed word set. \\
If \(\Sigma\) is a subset of \(\Omega\), a polynomial in words of \(\Sigma\) is a finite sum of products in the form \[a_0w_1(x_1)a_1w_2(x_2)a_2 \cdots \cdots w_n(x_n)a_n,\] where \(a_0, a_i \in Q\), \(w_i\in \Sigma\), \(x_i\in X\). Such a polynomial is called linear if it contains only one variable, say \(x\in X\), and is a finite sum of \(aw(x)b\), where \(a, b\in Q\) and \(w\in \Sigma\). The set of all linear polynomial in words of \(\Sigma\) is denoted by \(l(\Sigma)\).
In the main theorem of the paper under review it is proved that, if \(\Lambda\subseteq \Omega \setminus \Omega_m\), where \(m\geq 0\), is a semi-closed subset over \(\Omega_m\), then the following statements are equivalent:
\begin{enumerate}
\item There exist no monic linear identities in \(l(\Lambda \cup \Omega_m) \setminus l(\Omega_m)\).
\item There exist no linear identities in \(l(\Lambda \cup \Omega_m) \setminus l(\Omega_m)\).
\item The set \(\Lambda\) extends to a basis \(\Sigma\) of \(\Omega\), with \(\Sigma_m\) being a basis of \(\Omega_m\).
\item Any ordered basis \(\Sigma\) of \(\Omega\) has a subset \(\Theta\) with \(\Sigma_m \subseteq \Theta \subseteq \Sigma \setminus \Lambda\) such that the set \(\Theta \cup \Lambda\) forms a \(\Theta\)-ordered basis of \(\Omega\).
\end{enumerate}
Moreover, the set \(\Lambda\) is said to be independent over \(\Omega_m\) if one of the four equivalent conditions above is satisfied. Otherwise, \(\Lambda\) is said to be dependent.
Reviewer: Vincenzo De Filippis (Messina)Characterizations of automorphic and anti-automorphic involutions of the quaternionshttps://www.zbmath.org/1483.160422022-05-16T20:40:13.078697Z"Lawson, Jimmie"https://www.zbmath.org/authors/?q=ai:lawson.jimmie-d"Kizil, Eyüp"https://www.zbmath.org/authors/?q=ai:kizil.eyupIn the paper under review, the authors aim to characterize the automorphisms and anti-automorphisms of the division algebra of quaternions that are involutions. Two characterizations, one in terms of inner automorphisms and the other in terms of the involution eigenspaces, are given.
By the way, at the beginning of this paper, the authors state that ``After some preliminaries on quaternions in Section 2 and a brief review of connections with vector calculus in Section 3, we establish our main results in Section 4.''. However, this paper only consists of three sections: Section 1 Preliminaries, Section 2 Vector Calculus, and Section 3 Involutions.
Reviewer: Wu Jing (Fayetteville)Description of partial actionshttps://www.zbmath.org/1483.160432022-05-16T20:40:13.078697Z"Cortes, Wagner"https://www.zbmath.org/authors/?q=ai:cortes.wagner"Marcos, Eduardo N."https://www.zbmath.org/authors/?q=ai:marcos.eduardo-nSummary: In this paper, we study partial actions of groups on \(R\)-algebras, where \(R\) is a commutative ring. We describe the partial actions of groups on the indecomposable algebras with enveloping actions. Then we work on algebras that can be decomposed as product of indecomposable algebras and we give a description of the partial actions of groups on these algebras in terms of global actions.Principal Galois orders and Gelfand-Zeitlin moduleshttps://www.zbmath.org/1483.160442022-05-16T20:40:13.078697Z"Hartwig, Jonas T."https://www.zbmath.org/authors/?q=ai:hartwig.jonas-tSummary: We show that the ring of invariants in a skew monoid ring contains a so called standard Galois order. Any Galois ring contained in the standard Galois order is automatically itself a Galois order and we call such rings principal Galois orders. We give two applications. First, we obtain a simple sufficient criterion for a Galois ring to be a Galois order and hence for its Gelfand-Zeitlin subalgebra to be maximal commutative. Second, generalizing a recent result by Early-Mazorchuk-Vishnyakova, we construct canonical simple Gelfand-Zeitlin modules over any principal Galois order.
As an example, we introduce the notion of a rational Galois order, attached an arbitrary finite reflection group and a set of rational difference operators, and show that they are principal Galois orders. Building on results by Futorny-Molev-Ovsienko, we show that parabolic subalgebras of finite W-algebras are rational Galois orders. Similarly we show that Mazorchuk's orthogonal Gelfand-Zeitlin algebras of type \(A\), and their parabolic subalgebras, are rational Galois orders. Consequently we produce canonical simple Gelfand-Zeitlin modules for these algebras and prove that their Gelfand-Zeitlin subalgebras are maximal commutative.
Lastly, we show that quantum OGZ algebras, previously defined by the author, and their parabolic subalgebras, are principal Galois orders. This in particular proves the long-standing Mazorchuk-Turowska conjecture that, if \(q\) is not a root of unity, the Gelfand-Zeitlin subalgebra of \(U_q(\mathfrak{gl}_n)\) is maximal commutative and that its Gelfand-Zeitlin fibers are non-empty and (by Futorny-Ovsienko theory) finite.Product of generalized derivations with commuting values on a Lie idealhttps://www.zbmath.org/1483.160452022-05-16T20:40:13.078697Z"Carini, Luisa"https://www.zbmath.org/authors/?q=ai:carini.luisa"De Filippis, Vincenzo"https://www.zbmath.org/authors/?q=ai:de-filippis.vincenzo"Scudo, Giovanni"https://www.zbmath.org/authors/?q=ai:scudo.giovanniLet \(R\) be a prime ring with its right Martindale quotient ring \(Q\), \(Z(R)\) the center of \(R\), \(U\) the Utumi quotient ring of \(R\) and \(C = Z(U)\), the center of \(U\) (\(C\) is usually called the extended centroid of \(R\)). An additive map \(G : R \to R\) is called generalized derivation of \(R\) if there exists a derivation \(d\) of \(R\) such that \(G(xy) = G(x)y + xd(y)\), for all \(x, y \in R\).
\textit{M. Fošner} and \textit{J. Vukman} [Mediterr. J. Math. 9, No. 4, 847--863 (2012; Zbl 1262.16043)] proved that Let \(R\) be a prime ring of characteristic different from \(2\). If \(R\) admits generalized derivations \(F_1\) and \(F_2\) such that \(F_{1}(x)F_{2}(x) = 0\) for all that \(F_{1}(x) = xp\) and \(F_{2}(x) = qx\) for all \(x \in R\) and \(pq = 0\), except when at least one \(F_i\) is zero. \(x \in R\), then there exist \(p, q\) elements of the Martindale quotient ring \(Q\) of \(R\), such
In the paper under review, motivated by the above result, the authors investigate the case when the product of two generalized derivations is commuting on a Lie ideal \(L\) of \(R\), and prove the following theorem.
Theorem. Let \(R\) be a non-commutative prime ring of characteristic different from \(2\) with Utumi quotient ring \(U\) and extended centroid \(C\), \(L\) a non-central Lie ideal of \(R\), \(F\) and \(G\) two nonzero generalized derivations of \(R\). If \([F(u)G(u), u] = 0\) for all \(u \in L\), then one of the following holds:\begin{enumerate}
\item There exist \(u, v \in U\) such that \(uv \in C\) and \(F(x) = xu\), \(G(x) = vx\), for all \(x \in R\);
\item \(R \subseteq M_{2}(C)\).
\end{enumerate}
The result is obtained by using the theory of generalized polynomial identities [\textit{K. I. Beidar} et al., Rings with generalized identities. New York, NY: Marcel Dekker (1996; Zbl 0847.16001)] and the theory of differential identities [\textit{V. K. Kharchenko}, Algebra i Logika 17, No. 2, 220--238, 242--243 (1978)].
For the entire collection see [Zbl 1455.53022].
Reviewer: Nadeem ur Rehman (Aligarh)Jordan higher derivations of incidence algebrashttps://www.zbmath.org/1483.160462022-05-16T20:40:13.078697Z"Chen, Lizhen"https://www.zbmath.org/authors/?q=ai:chen.lizhen"Xiao, Zhankui"https://www.zbmath.org/authors/?q=ai:xiao.zhankuiSummary: Let \(\mathcal{R}\) be a 2-torsion-free commutative ring with unity and \(X\) be a locally finite pre-ordered set. We prove in this paper that every Jordan higher derivation on the incidence algebra \(I(X,\mathcal{R})\) is a higher derivation. By the way, we also provide a new proof of the known fact that every Jordan derivation of \(I(X,\mathcal{R})\) is a derivation.Extensions and deformations of algebras with higher derivationshttps://www.zbmath.org/1483.160472022-05-16T20:40:13.078697Z"Das, Apurba"https://www.zbmath.org/authors/?q=ai:das.apurba|das.apurba.1Summary: Higher derivations on an associative algebra generalize higher-order derivatives. We call a tuple consisting of an algebra and a higher derivation on it by an AssHDer pair. We define cohomology for AssHDer pairs with coefficients in a representation. Next, we study central extensions of an AssHDer pair and relate them with the second cohomology group of the AssHDer pair. Finally, we consider deformations of AssHDer pairs that are governed by the cohomology with self-coefficient.Product of generalized skew derivations on Lie idealshttps://www.zbmath.org/1483.160482022-05-16T20:40:13.078697Z"De Filippis, Vincenzo"https://www.zbmath.org/authors/?q=ai:de-filippis.vincenzoMany authors in literature studied the set \(P(G,S)=\{G(x)x:x\in S\}\), where \(S\) is an appropriate subset of a prime ring \(R\) and \(G\) an additive map defined on \(R\). Since the studies on \(P(G,S)\) imply that it is rather large in \(R\) when \(G\) is a non-zero generalized derivation, recently many papers consider a prime ring \(R\) that admits an additive map \(F\) satisfying the following condition:
\[
F(x)=0 \ \text{ for all }x\in P(G,S) \tag{1}
\]
In this paper, the author follows this line of investigation: he considers \(R\) a prime ring, with \(\mathrm{char}(R)\neq2\), satisfying (1), when \(F\) and \(G\) are both non-zero generalized skew derivations of \(R\), associated with the same automorphism \(\alpha\), and \(S\) a non-central Lie ideal of \(R\). He proves that, if \(Q_r\) is the right Martindale quotient ring of \(R\) and \(C\) its extended centroid, then one of the following holds:
\begin{itemize}
\item[1.] there exist \(a,c\in Q_r\) such that \(F(x)=ax\) and \(G(x)=cx\), for all \(x\in R\), with \(ac=0\);
\item[2.] \(R\subseteq M_2(C)\), the ring of all \(2\times2\) matrices over \(C\), and there exist \(a,b,c,q\in Q_r\), with \(q\) invertible element of \(Q_r\), such that \(F(x)=ax+qxq^{-1}b\) and \(G(x)=cx\), for all \(x\in R\), with \(ac+qcq^{-1}b=0\);
\item[3.] \(R\subseteq M_2(C)\) and there exist \(a,b,c\in Q_r\) such that \(F(x)=ax+\alpha(x)b\) and \(G(x)=cx\), for all \(x\in R\), with \(ac=\alpha(c)b=0\). Moreover, in this case \(\alpha\) is not an inner automorphism of \(R\).
\end{itemize}
Reviewer: Giovanni Scudo (Messina)Dhara-Rehman-Raza's identities on left ideals of prime ringshttps://www.zbmath.org/1483.160492022-05-16T20:40:13.078697Z"Fahid, Brahim"https://www.zbmath.org/authors/?q=ai:fahid.brahim"Bennis, Driss"https://www.zbmath.org/authors/?q=ai:bennis.driss"Mamouni, Abdellah"https://www.zbmath.org/authors/?q=ai:mamouni.abdellahIn this manuscript, the authors examined the commutativity of a prime ring \(R\) satisfying some differential identities on a suitable subset of \(R\). The authors have mainly investigated the results obtained by \textit{B. Dhara} et al. [Miskolc Math. Notes 16, No. 2, 769--779 (2016; Zbl 1349.16068)] on a nonzero left ideal of \(R\).
Reviewer: Mohd Arif Raza (Rabigh)Zariski topologies on graded idealshttps://www.zbmath.org/1483.160502022-05-16T20:40:13.078697Z"Bataineh, Malik"https://www.zbmath.org/authors/?q=ai:bataineh.malik"Alshehry, Azzh Saad"https://www.zbmath.org/authors/?q=ai:alshehry.azzh-saad"Abu-Dawwas, Rashid"https://www.zbmath.org/authors/?q=ai:abu-dawwas.rashidSummary: In this paper, we show there are strong relations between the algebraic properties of a graded commutative ring \(R\) and topological properties of open subsets of Zariski topology on the graded prime spectrum of \(R\). We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in \(R\) by using topological properties.On the structure of zero-divisor elements in a near-ring of skew formal power serieshttps://www.zbmath.org/1483.160512022-05-16T20:40:13.078697Z"Alhevaz, Abdollah"https://www.zbmath.org/authors/?q=ai:alhevaz.abdollah"Hashemi, Ebrahim"https://www.zbmath.org/authors/?q=ai:hashemi.ebrahim"Shokuhifar, Fatemeh"https://www.zbmath.org/authors/?q=ai:shokuhifar.fatemehLet \(\alpha \) be an endomorphism of the unital ring \(R\) with \( R[[x;\alpha ]]\) the skew power series ring with indeterminate \(x.\) The subset of all power series with positive order is denoted by \( R_{0}[[x;\alpha ]]\) and is a (left) near-ring with respect to addition and composition. The set of zero-divisors of a ring \(R\) is denoted by \(Z(R)\).
The authors study the zero-divisors of the near-ring \( R_{0}[[x;\alpha ]].\) At the outset, such elements are characterized in terms of the structural properties of the near-ring \(R_{0}[[x;\alpha ]].\) An important and useful result to advance their investigations is the following: For \(f=\sum\limits_{i=1}^{\infty }a_{i}x^{i}\) and \( g=\sum\limits_{i=1}^{\infty }b_{i}x^{i}\) two non-zero elements of \( R_{0}[[x;\alpha ]]\) with \(f\circ g=0,\) it follows that (i) \(a_{1}b_{1}=0;\) (ii) \(rf=0\) for some non-zero \(r\in R\) and (iii), \(f\) is nilpotent or \(sg=0\) for some non-zero \(s\in R\). Conditions when \(Z(R_{0}[[x;\alpha ]]\)) is an ideal of \(R_{0}[[x;\alpha ]]\) are determined. The zero-divisor graph of the near-ring \(R_{0}[[x;\alpha ]]\) is the (undirected) graph with vertices the non-zero zero-divisors of \(R_{0}[[x;\alpha ]]\) and two distinct vertices \(f\) and \(g\) are adjacent if and only if \(f\circ g=0\) or \(g\circ f=0.\) It is shown that the diameter of this graph is either \(2\) or \(3\).
Reviewer: Stefan Veldsman (Port Elizabeth)Study of ring structure from multiset contexthttps://www.zbmath.org/1483.160522022-05-16T20:40:13.078697Z"Debnath, Shyamal"https://www.zbmath.org/authors/?q=ai:debnath.shyamal"Debnath, Amaresh"https://www.zbmath.org/authors/?q=ai:debnath.amareshSummary: The concept of multiset is a generalization of the Cantor set. In this paper we have introduced the notion of multirings (in short mrings) and study some of their important properties. It is shown that the intersection of two mrings is again a mring but their union may not be a mring and a mring over a non-commutative ring may be commutative.From Schouten to Mackenzie: notes on bracketshttps://www.zbmath.org/1483.170012022-05-16T20:40:13.078697Z"Kosmann-Schwarzbach, Yvette"https://www.zbmath.org/authors/?q=ai:kosmann-schwarzbach.yvetteSummary: In this paper, dedicated to the memory of Kirill Mackenzie, I relate the origins and early development of the theory of graded Lie brackets, first in the publications on differential geometry of Schouten, Nijenhuis, and Frölicher-Nijenhuis, then in the work of Gerstenhaber and Nijenhuis-Richardson in cohomology theory.Homological invariants of the arrow removal operationhttps://www.zbmath.org/1483.180112022-05-16T20:40:13.078697Z"Erdmann, Karin"https://www.zbmath.org/authors/?q=ai:erdmann.karin"Psaroudakis, Chrysostomos"https://www.zbmath.org/authors/?q=ai:psaroudakis.chrysostomos"Solberg, Øyvind"https://www.zbmath.org/authors/?q=ai:solberg.oyvindSummary: In this paper we show that Gorensteinness, singularity categories and the finite generation condition \(\mathsf{Fg}\) for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra.Extension groups between atoms in abelian categorieshttps://www.zbmath.org/1483.180122022-05-16T20:40:13.078697Z"Kanda, Ryo"https://www.zbmath.org/authors/?q=ai:kanda.ryoIn [J. Algebra 592, 233--299 (2022; Zbl 1481.18011)], the author introduced the atom spectrum of an abelian category which can be regarded as a generalization of the set of prime ideals of a commutative ring. Now, he defines extension groups between atoms and uses them to characterize localizing subcategories closed under injective envelopes of a locally noetherian Grothendieck category. Since localizing subcategories in the category of modules over a commutative noetherian ring are closed under injective envelopes, this generalizes the well-known characterization of localizing subcategories in the category of modules over a commutative noetherian ring.
Reviewer: Jiří Rosický (Brno)Some characterizations of Auslander and Bass classeshttps://www.zbmath.org/1483.180162022-05-16T20:40:13.078697Z"Huang, Yuntao"https://www.zbmath.org/authors/?q=ai:huang.yuntao"Song, Weiling"https://www.zbmath.org/authors/?q=ai:song.weilingSummary: Let \(R\) and \(S\) be rings and \(_RC_S\) a semidualizing bimodule. For a subcategory \(\mathcal{X}\) of the Auslander class \(\mathcal{A}_C(S)\) containing all projective and \(C\)-injective modules, we show that a module \(N\in \mathcal{A}_C(S)\) if and only if there exists an exact sequence \(\cdots \rightarrow X_i\rightarrow \cdots \rightarrow X_1\rightarrow X_0\rightarrow X^0\rightarrow X^1\rightarrow \cdots \rightarrow X^i\rightarrow \cdots\) in \(\operatorname{Mod} S\) with all \(X_i,X^i\) in \(\mathcal{X}\) such that it remains exact after applying the functor \(\operatorname{Hom}_S(-,E)\) for any \(C\)-injective module \(E\) and \(N\cong \operatorname{Im} (X_0\rightarrow X^0)\). For a subcategory \(\mathcal{Y}\) of the Bass class \(\mathcal{B}_C(R)\) containing all injective and \(C\)-projective modules, we show that a module \(M\in \mathcal{B}_C(R)\) if and only if there exists an exact sequence \(\cdots \rightarrow Y_i\rightarrow \cdots \rightarrow Y_1\rightarrow Y_0\rightarrow Y^0\rightarrow Y^1\rightarrow \cdots \rightarrow Y^i\rightarrow \cdots\) in \(\operatorname{Mod} R\) with all \(Y_i,Y^i\) in \(\mathcal{Y}\) such that it remains exact after applying the functor \(\operatorname{Hom}_S(Q,-)\) for any \(C\)-projective module \(Q\) and \(M\cong \operatorname{Im}(Y_0\rightarrow Y^0)\). We apply these results to comparison of some relative homological dimensions.Relative rigid objects in extriangulated categorieshttps://www.zbmath.org/1483.180182022-05-16T20:40:13.078697Z"Liu, Yu"https://www.zbmath.org/authors/?q=ai:liu.yu|liu.yu.1|liu.yu.2"Zhou, Panyue"https://www.zbmath.org/authors/?q=ai:zhou.panyueThe authors study the relation between relative cluster tilting theory in extriangulated categories and \(\tau\)-tilting theory in module categories, generalizing results of Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. First, they study properties of \(R\)-rigid objects over a Krull-Schmidt, Hom-finite, extriangulated category. Second, gives a bijection between certain isomorphisms classes of basic \(R\)-rigid objects and classes of basic \(\tau\)-rigid pairs of modules (Thm 3.13); which allows to give an equivalent characterization on tilting modules (Thm 3.14). Third, study the relationship between \(R\)-rigid, rigid and d-rigid (Thm 3.17). Finally, with an example, they illustrated the main results.
Reviewer: Luz Adriana Mejia Castaño (Barranquilla)PBW property for associative universal enveloping algebras over an operadhttps://www.zbmath.org/1483.180252022-05-16T20:40:13.078697Z"Khoroshkin, Anton"https://www.zbmath.org/authors/?q=ai:khoroshkin.antonIf \(P\) is an operad and \(V\) is a \(P\)-algebra, the category of \(V\)-modules is isomorphic to the category of modules over an algebra called the \(P\)-enveloping algebra of \(P\). The Poincaré-Birkhoff-Witt (PBW) property of this object is studied. When \(P\) is Koszul, a criterion for PBW property is given. A necessary condition on the Hilbert series of \(P\) is given, and a sufficient condition is given for operads with a Gröbner basis in term of the structure of the underlying trees. As an application it is shown that the operads of Lie, commutative associative, associative, compatible Lie, pre-Lie, Zinbiel algebras have the PBW property, whereas the operads of Poisson, permutative, Leibniz algebras have not.
Reviewer: Loïc Foissy (Calais)Hopf-Galois structures on finite extensions with quasisimple Galois grouphttps://www.zbmath.org/1483.200052022-05-16T20:40:13.078697Z"Tsang, Cindy (Sin Yi)"https://www.zbmath.org/authors/?q=ai:tsang.cindy-sin-yiThis is an interesting paper about determining all possible Hopf Galois structures on a given \(G\)-Galois extension \(L/K\) of fields. Often, there are very many such structures. On the other hand, \textit{N. P. Byott} [Bull. Lond. Math. Soc. 36, No. 1, 23--29 (2004; Zbl 1038.12002)] proved that if \(G\) is simple nonabelian, there are exactly two Hopf Galois structures on \(L\), the classical one, and another one which is explicitly known and has interesting properties. The proof uses some results stemming from CFSG (the classification of finite simple groups) in a crucial way. The paper under review generalizes Byott's result, with literally the same outcome, to the class of quasisimple groups \(G\). A group \(G\) is quasisimple if \(G\) has no nontrivial abelian quotient and the factor group of \(G\) modulo its center is simple nonabelian. As examples, the paper mentions so-called double covers of the alternating group \(A_n\). Perhaps more accessible to non-specialists are the examples \(\mathrm{SL}_n(\mathbb F_q)\) for \(n\ge 2\) (for \(n=2\) one needs to exclude \(q=2,3\)). Again, the main effort in the proof of this result comes from group theory, and CFSG in particular. It is not enough to simply quote some of the spinoffs of CFSG; a lot of detailed work is required to go through a well-organised series of steps, excluding more and more possibilities, until in the end only two options for the Hopf Galois structure remain in play. This is done very cleverly. One central notion is the theory of Schur multipliers. For the details we have to refer to the paper. The ingredients coming from CFSG are neatly assembled in a short extra section, before the hard work begins. Some of these ingredients were already used in the above-mentioned paper of Byott. To give the reader a flavor of these ingredients, let us mention just one of them, due to Guralnick: Having a subgroup of prime power index is a severe restriction on a simple group \(A\). There is a 5-item list of simple groups \(A\) having such subgroups, two items consisting of explicit series, and the other items covering just four groups, two of which are Mathieu groups, and thus belong to the finite list of the sporadic simple groups. The paper is nicely rounded off by the information (see p.149) that the results will not carry over as they stand to other variants of the notion ``simple non-abelian'', like almost simple or characteristically simple groups.
Reviewer: Cornelius Greither (Neubiberg)On the structure of the augmentation quotient groups for some nonabelian groupshttps://www.zbmath.org/1483.200062022-05-16T20:40:13.078697Z"Zhao, Hongmei"https://www.zbmath.org/authors/?q=ai:zhao.hongmei"Tang, Guoping"https://www.zbmath.org/authors/?q=ai:tang.guoping(no abstract)The \(L^2\)-torsion polytope of amenable groupshttps://www.zbmath.org/1483.200782022-05-16T20:40:13.078697Z"Funke, Florian"https://www.zbmath.org/authors/?q=ai:funke.florianSummary: We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the \(L^2\)-torsion polytope among \(G\)-CW-complexes for these groups. As another application we prove that the \(L^2\)-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.Cellularity of endomorphism algebras of Young permutation moduleshttps://www.zbmath.org/1483.200862022-05-16T20:40:13.078697Z"Donkin, Stephen"https://www.zbmath.org/authors/?q=ai:donkin.stephenLet \(E\) be an \(n\)-dimensional vector space. This paper studies the question on whether the endomorphism algebra End\(_{\mathrm{Sym}(n)}(E^{\otimes r})\) is cellular, where the symmetric group \(\mathrm{Sym}(n)\) acts on \(E\) by permuting the elements of a basis and hence on the \(E^{\otimes r}\). The author shows that the endomorphism algebra of the permutation module on an arbitrary Young \(\mathrm{Sym}(n)\)-set is a cellular algebra, and determines, in terms of the point stabilisers which appear, when the endomorphism algebra is quasi-hereditary.
Reviewer: Hu Jun (Beijing)Stratification and \(\pi\)-cosupport: finite groupshttps://www.zbmath.org/1483.200882022-05-16T20:40:13.078697Z"Benson, Dave"https://www.zbmath.org/authors/?q=ai:benson.david-john"Iyengar, Srikanth B."https://www.zbmath.org/authors/?q=ai:iyengar.srikanth-b"Krause, Henning"https://www.zbmath.org/authors/?q=ai:krause.henning"Pevtsova, Julia"https://www.zbmath.org/authors/?q=ai:pevtsova.juliaSummary: We introduce the notion of \(\pi \)-cosupport as a new tool for the stable module category of a finite group scheme. In the case of a finite group, we use this to give a new proof of the classification of tensor ideal localising subcategories. In a sequel to this paper, we carry out the corresponding classification for finite group schemes.Rings arising from finite tight hypergroupshttps://www.zbmath.org/1483.201212022-05-16T20:40:13.078697Z"French, Christopher"https://www.zbmath.org/authors/?q=ai:french.christopher-p"Zieschang, Paul-Hermann"https://www.zbmath.org/authors/?q=ai:zieschang.paul-hermannThe notion of a hypergroup (in the sense of Marty) provides a far reaching and meaningful generalization of the concept of a group. Specific classes of hypergroups have given rise to challenging questions and interesting connections to geometric and group theoretic topics. The present article is a continuation of a study of residually thin hypergroups which was initiated in [\textit{C. French} and \textit{P.-H. Zieschang}, J. Algebra 551, 93--118 (2020; Zbl 07175321)]. One of the main results of [loc. cit.] says that finite tight hypergroups, that is hypergroups all elements \(h\) of which satisfy \(hh^*h = \{h\}\), are residually thin. In this paper, the authors take advantage of this result in order to construct, for each finite tight hypergroup satisfying a mild extra condition, an associative ring. The rings which they found generalize the construction of scheme rings for a certain class of association schemes, including those which correspond to finite groups (group rings).
Remark: The hypergroup that the authors study in this paper (a special case of hypergroups) is usually called polygroup, see [the reviewer, Polygroup theory and related systems. Hackensack, NJ: World Scientific (2013; Zbl 1266.20083)].
Reviewer: Bijan Davvaz (Yazd)On locally \(A\)-convex moduleshttps://www.zbmath.org/1483.460502022-05-16T20:40:13.078697Z"Haralampidou, M."https://www.zbmath.org/authors/?q=ai:haralampidou.marina"Oudadess, M."https://www.zbmath.org/authors/?q=ai:oudadess.mohamed"Palacios, L."https://www.zbmath.org/authors/?q=ai:palacios.lourdes"Signoret, C."https://www.zbmath.org/authors/?q=ai:signoret.carlos-j-eThe paper deals with the inheritance of the property of being some type of an \(\mathcal A\)-module for a locally convex algebra \(({\mathcal A}, (\mid\!\cdot\!\mid_i)_{i\in I})\). More precisely, the authors are interested in the following classes of \(\mathcal A\)-modules:\par Let \((E, (\mid\!\cdot\!\mid_\lambda)_{\lambda\in\Lambda})\) be a left \(\mathcal A\)-module for a locally convex algebra \(({\mathcal A}, (\mid\!\cdot\!\mid_i)_{i\in I})\). Then \(E\) is\par a) a left \(\mathcal A\)-module with a jointly continuous action if, for each \(\lambda\in \Lambda\), there exist \(i(\lambda)\in I\) and \(\lambda'\in\Lambda\) such that \(\mid ax\!\mid_\lambda\leqslant\mid \!a\!\mid_{i(\lambda)}\mid\!x\mid_{\lambda'}\) for every \(a\in\mathcal A\) and \(x\in E\);\par b) a locally \(A\)-convex \(\mathcal A\)-module if, for each \(a\in\mathcal A\) and each \(\lambda\in\Lambda\), there exists \(\alpha(a, \lambda)>0\) such that \(\mid ax\!\mid_\lambda\leqslant\alpha(a, \lambda)\mid\! x\mid_\lambda\) for every \(x\in E\);\par c) a locally uniformly \(A\)-convex left \(\mathcal A\)-module if, for each \(a\in\mathcal A\), there exists \(\alpha(a)>0\) such that \(\mid ax\!\mid_\lambda\leqslant\alpha(a)\mid\! x\mid_\lambda\) for every \(\lambda\in\Lambda\) and \(x\in E\);\par d) a locally \(m\)-convex left \(\mathcal A\)-module if, for each \(\lambda\in\Lambda\), there exists \(i(\lambda)\in I\) such that \(\mid ax\!\mid_\lambda\leqslant\mid\!a\!\mid_{i(\lambda)}\mid\! x\mid_\lambda\) for every \(a\in\mathcal A\) and \(x\in E\).\par Many examples of algebras, belonging to the one of the classes and not in the another, are provided.\par The authors show that the property of being locally \(A\)-convex left \(\mathcal A\)-module is inherited by the operations of\par (i) taking the quotient by a left \(\mathcal A\)-submodule;\par (ii) considering instead of a non-unital algebra \(\mathcal A\) its unitization;\par (iii) taking a completion of a locally \(A\)-convex left \(\mathcal A\)-module;\par (iv) taking the (direct) product left \(\mathcal A\)-module of locally \(A\)-convex left \(\mathcal A\)-modules;\par (v) taking the projective limit of a projective system of locally \(A\)-convex left \(\mathcal A\)-modules;\par and that the property of being locally \(m\)-convex left \(\mathcal A\)-module is inherited by the operation of\par (vi) taking the strict inductive limit of a sequence of locally \(m\)-convex left \(\mathcal A\)-modules.\par The authors also claim that the properties (i)-(v) are also true for left \(\mathcal A\)-modules with jointly continuous action, locally uniformly \(A\)-convex left \(\mathcal A\)-modules and locally \(m\)-convex left \(\mathcal A\)-modules.\par At the end of the paper, the authors offer some Arens-Michael-like decomposition for some projective sytems of \(m\)-normed or \(A\)-normed left \(\mathcal A\)-modules.
Reviewer: Mart Abel (Tartu)Characterizing Jordan homomorphismshttps://www.zbmath.org/1483.470722022-05-16T20:40:13.078697Z"Mathieu, Martin"https://www.zbmath.org/authors/?q=ai:mathieu.martinSummary: It is shown that every bounded, unital linear mapping that preserves elements of square zero from a \(C^*\)-algebra of real rank zero and without tracial states into a Banach algebra is a Jordan homomorphism.The Kontsevich integral for bottom tangles in handlebodieshttps://www.zbmath.org/1483.570122022-05-16T20:40:13.078697Z"Habiro, Kazuo"https://www.zbmath.org/authors/?q=ai:habiro.kazuo"Massuyeau, Gwénaël"https://www.zbmath.org/authors/?q=ai:massuyeau.gwenaelThe Kontsevich integral was initially defined for links in the 3-sphere [\textit{D. Bar-Natan}, Topology 34, No. 2, 423--472 (1995; Zbl 0898.57001)] and was then extended to tangles in the 3-ball [\textit{T. Q. T. Le} and \textit{J. Murakami}, Compos. Math. 102, No. 1, 41--64 (1996; Zbl 0851.57007)]. In the paper under review, the authors define a Kontsevich integral for tangles in handlebodies. For this, they consider a suitable category of bottom tangles in handlebodies and a category of Jacobi diagrams in handlebodies. Their generalized Kontsevich integral is defined as a functor between these categories. A detailed description of the two categories is given and the Kontsevich invariant is computed on the generators of the source category. The authors prove that their invariant is a universal finite type invariant, and that it refines the LMO functor -- a functorial version of the Le-Murakami-Ohtsuki invariant -- of special Lagrangian cobordisms.
Reviewer: Delphine Moussard (Marseille)Convex algebras of probability distributions induced by finite associative ringshttps://www.zbmath.org/1483.600032022-05-16T20:40:13.078697Z"Yashunsky, Alexey D."https://www.zbmath.org/authors/?q=ai:yashunsky.aleksey-dSummary: We consider the transformations of random variables over a finite associative ring by the addition and multiplication operations. For arbitrary finite rings, we construct families of distribution algebras, which are sets of distributions closed over sums and products of independent random variables.The triple-pair construction for weighted \(\omega\)-pushdown automatahttps://www.zbmath.org/1483.681632022-05-16T20:40:13.078697Z"Droste, Manfred"https://www.zbmath.org/authors/?q=ai:droste.manfred"Ésik, Zoltán"https://www.zbmath.org/authors/?q=ai:esik.zoltan"Kuich, Werner"https://www.zbmath.org/authors/?q=ai:kuich.wernerSummary: Let \(S\) be a complete star-omega semiring and \(\Sigma\) be an alphabet. For a weighted \(\omega\)-pushdown automaton \(\mathscr{P}\) with stateset \(\{1, \dots, n\}\), \(n\geq 1\), we show that there exists a mixed algebraic system over a complete semiring-semimodule pair (\((S\ll\Sigma^*\gg )^{n\times n}\), \((S\ll\Sigma^\omega\gg)^n\)) such that the behavior \(\|\mathscr{P}\|\) of \(\mathscr{P}\) is a component of a solution of this system. In case the basic semiring is \(\mathbb{B}\) or \(\mathbb{N}^\infty\), we show that there exists a mixed context-free grammar that generates \(\|\mathscr{P}\|\). The construction of the mixed context-free grammar from \(\mathscr{P}\) is a generalization of the well known triple construction and is called now triple-pair construction for \(\omega\)-pushdown automata.
For the entire collection see [Zbl 1433.68013].On primary ideals. Ihttps://www.zbmath.org/1483.684952022-05-16T20:40:13.078697Z"Watase, Yasushige"https://www.zbmath.org/authors/?q=ai:watase.yasushigeSummary: We formalize in the Mizar System [\textit{G. Bancerek} et al., Lect. Notes Comput. Sci. 9150, 261--279 (2015; Zbl 1417.68201); J. Autom. Reasoning 61, No. 1--4, 9--32 (2018; Zbl 1433.68530)], definitions and basic propositions about primary ideals of a commutative ring along with Chapter 4 of [\textit{M. F. Atiyah} and \textit{I. G. Macdonald}, Introduction to commutative algebra. Reading, Mass.-Menlo Park, Calif.-London-Don Mills, Ont.: Addison-Wesley Publishing Company (1969; Zbl 0175.03601)] and Chapter III of [\textit{O. Zariski} and \textit{P. Samuel}, Commutative Algebra. Vol. 1. 2nd ed. New York - Heidelberg - Berlin: Springer-Verlag (1975; Zbl 0313.13001)]. Additionally other necessary basic ideal operations such as compatibilities taking radical and intersection of finite number of ideals are formalized as well in order to prove theorems relating primary ideals. These basic operations are mainly quoted from Chapter 1 of [Atiyah and Macdonald, loc. cit.] and compiled as preliminaries in the first half of the article.Generalized multifractality at spin quantum Hall transitionhttps://www.zbmath.org/1483.810732022-05-16T20:40:13.078697Z"Karcher, Jonas F."https://www.zbmath.org/authors/?q=ai:karcher.jonas-f"Charles, Noah"https://www.zbmath.org/authors/?q=ai:charles.noah"Gruzberg, Ilya A."https://www.zbmath.org/authors/?q=ai:gruzberg.ilya-a"Mirlin, Alexander D."https://www.zbmath.org/authors/?q=ai:mirlin.aleksandr-davidovichSummary: Generalized multifractality characterizes scaling of eigenstate observables at Anderson-localization critical points. We explore generalized multifractality in 2D systems, with the main focus on the spin quantum Hall (SQH) transition in superconductors of symmetry class C.
Relations and differences with the conventional integer quantum Hall (IQH) transition are also studied. Using the field-theoretical formalism of non-linear sigma-model, we derive the pure-scaling operators representing generalizing multifractality and then ``translate'' them to the language of eigenstate observables. Performing numerical simulations on network models for SQH and IQH transitions, we confirm the analytical predictions for scaling observables and determine the corresponding exponents. Remarkably, the generalized-multifractality exponents at the SQH critical point strongly violate the generalized parabolicity of the spectrum, which implies violation of the local conformal invariance at this critical point.Brief lectures on duality, integrability and deformationshttps://www.zbmath.org/1483.810852022-05-16T20:40:13.078697Z"Klimčík, Ctirad"https://www.zbmath.org/authors/?q=ai:klimcik.ctiradGroup theoretical derivation of consistent massless particle theorieshttps://www.zbmath.org/1483.810872022-05-16T20:40:13.078697Z"Nisticò, Giuseppe"https://www.zbmath.org/authors/?q=ai:nistico.giuseppeSummary: Current theories of massless free particle assume \textit{unitary} space inversion and \textit{anti-unitary} time reversal operators. In so doing robust classes of possible theories are discarded. In the present work theories of massless systems are derived through a strictly deductive development from the principle of relativistic invariance, so that a kind of space inversion or time reversal operator is ruled out only if it causes inconsistencies. As results, new classes of consistent theories for massless isolated systems are explicitly determined. On the other hand, the approach determines definite constraints implied by the invariance principle; they were ignored by some past investigations that, as a consequence, turn out to be not consistent with the invariance principle. Also the problem of the \textit{localizability} for massless systems is reconsidered within the new theoretical framework, obtaining a generalization and a deeper detailing of previous results.Finite-\(N\) corrections to the superconformal index of toric quiver gauge theorieshttps://www.zbmath.org/1483.811232022-05-16T20:40:13.078697Z"Arai, Reona"https://www.zbmath.org/authors/?q=ai:arai.reona"Fujiwara, Shota"https://www.zbmath.org/authors/?q=ai:fujiwara.shota"Imamura, Yosuke"https://www.zbmath.org/authors/?q=ai:imamura.yosuke"Mori, Tatsuya"https://www.zbmath.org/authors/?q=ai:mori.tatsuyaSummary: The superconformal index of quiver gauge theories realized on D3-branes in toric Calabi-Yau cones is investigated. We use the AdS/CFT correspondence and study D3-branes wrapped on supersymmetric cycles. We focus on brane configurations in which a single D3-brane is wrapped on a cycle, and we do not take account of branes with multiple wrapping. We propose a formula that gives finite-\(N\) corrections to the index caused by such brane configurations. We compare the predictions of the formula for several examples with the results on the gauge theory side obtained by using localization for small sizes of gauge groups, and confirm that the formula correctly reproduces the finite-\(N\) corrections up to the expected order.Center group dominance in quark confinementhttps://www.zbmath.org/1483.811472022-05-16T20:40:13.078697Z"Ikeda, Ryu"https://www.zbmath.org/authors/?q=ai:ikeda.ryu"Kondo, Kei-Ichi"https://www.zbmath.org/authors/?q=ai:kondo.kei-ichiSummary: We show that the color \(N\)-dependent area law falloffs of the double-winding Wilson loop averages for the \(SU(N)\) lattice gauge theory obtained in previous works are reproduced from the corresponding lattice abelian gauge theory with the center gauge group \(Z_N\). This result indicates the center group dominance in quark confinement.Hilbert-space fragmentation, multifractality, and many-body localizationhttps://www.zbmath.org/1483.811682022-05-16T20:40:13.078697Z"Pietracaprina, Francesca"https://www.zbmath.org/authors/?q=ai:pietracaprina.francesca"Laflorencie, Nicolas"https://www.zbmath.org/authors/?q=ai:laflorencie.nicolasSummary: Investigating many-body localization (MBL) using exact numerical methods is limited by the exponential growth of the Hilbert space. However, localized eigenstates display multifractality and only extend over a vanishing fraction of the Hilbert space.
Here, building on this remarkable property, we develop a simple yet efficient decimation scheme to discard the irrelevant parts of the Hilbert space of the random-field Heisenberg chain. This leads to a Hilbert space fragmentation in small clusters, allowing to access larger systems at strong disorder. The MBL transition is quantitatively predicted, together with a geometrical interpretation of MBL multifractality as a shattering of the Hilbert space.The microscopic picture of the integer quantum Hall regimehttps://www.zbmath.org/1483.811702022-05-16T20:40:13.078697Z"Römer, Rudolf A."https://www.zbmath.org/authors/?q=ai:romer.rudolf-a"Oswald, Josef"https://www.zbmath.org/authors/?q=ai:oswald.josefSummary: Computer modeling of the integer quantum Hall effect based on self-consistent Hartree-Fock (HF) calculations has now reached an astonishing level of maturity. Spatially-resolved studies of the electron density at near macroscopic system sizes of up to \(1\mu\text{m}^2\) reveal self-organized clusters of locally fully filled and locally fully depleted Landau levels depending on which spin polarization is favored. The behavior results, for strong disorders, in an exchange-interaction induced \(g\)-factor enhancement and, ultimately, gives rise to narrow transport channels, including the celebrated narrow edge channels. For weak disorder, we find that bubble and stripes phases emerge with characteristics that predict experimental results very well. Hence the HF approach has become a convenient numerical basis to \textit{quantitatively} study the quantum Hall effects, superseding previous more qualitative approaches.5D \(\mathcal{N} = 1\) super QFT: symplectic quivershttps://www.zbmath.org/1483.830262022-05-16T20:40:13.078697Z"Saidi, E. H."https://www.zbmath.org/authors/?q=ai:saidi.el-hassan"Drissi, L. B."https://www.zbmath.org/authors/?q=ai:drissi.lalla-btissamSummary: We develop a method to build new 5D \(\mathcal{N} = 1\) gauge models based on Sasaki-Einstein manifolds \(Y^{p, q}\). These models extend the standard 5D ones having a unitary \(\mathrm{SU}(p)_q\) gauge symmetry based on \(Y^{p, q} \). Particular focus is put on the building of a gauge family with symplectic \(\mathrm{SP}(2r, \mathbb{R})\) symmetry. These super QFTs are embedded in M-theory compactified on folded toric Calabi-Yau threefolds \(\hat{X}(Y^{2r, 0})\) constructed from conical \(Y^{2r, 0}\). By using outer-automorphism symmetries of 5D \(\mathcal{N} = 1\) BPS quivers with unitary \(\mathrm{SU}(2r)\) gauge invariance, we also construct BPS quivers with symplectic \(\mathrm{SP}(2r, \mathbb{R})\) gauge symmetry. Other related aspects are discussed.Electrostatic description of five-dimensional SCFTshttps://www.zbmath.org/1483.830812022-05-16T20:40:13.078697Z"Legramandi, Andrea"https://www.zbmath.org/authors/?q=ai:legramandi.andrea"Nunez, Carlos"https://www.zbmath.org/authors/?q=ai:nunez.carlosSummary: In this paper we discuss an infinite class of \(\mathrm{AdS}_6\) backgrounds in Type IIB supergravity dual to five dimensional SCFTs whose low energy description is in terms of linear quiver theories. The quantisation of the Page charges imposes that each solution is determined once a convex, piece-wise linear function is specified. In the dual field theory, we interpret this function as encoding the ranks of colour and flavour groups in the associated quiver. We check our proposal with several examples and provide general expressions for the holographic central charge and the Wilson loop VEV. Some solutions outside this general class, with less clear quiver interpretation, are also discussed.