Recent zbMATH articles in MSC 16https://www.zbmath.org/atom/cc/162021-04-16T16:22:00+00:00WerkzeugOn locally solvable subgroups in division rings.https://www.zbmath.org/1456.200322021-04-16T16:22:00+00:00"Huỳnh Vi\d{ê}t Khánh"https://www.zbmath.org/authors/?q=ai:huynh-viet-khanh.Let $D$ be division ring with centre $F$. The most striking result of this paper is that if $G$ is a locally soluble subnormal subgroup of $D^*$ such that the $i$-th member $G^{(i)}$ of the derived series of $G$, for some finite $i$, is algebraic over $F$, then $G$ is central in $D$. Various special cases of this are known and are listed in this paper.
The author then widens his study. Suppose $M$ is a non-abelian, locally soluble maximal subgroup of the subnormal subgroup $G$ of $D^*$ such that $M^{(i)}$ is algebraic over $F$, again with $i$ finite. Then $D$ is a cyclic algebra of prime degree. In Theorem 2.7, further information is given concerning the structure of $G$. For example, the FC-centre of $G$ and the Fitting subgroup of $M$ are equal and are described in terms of the cyclic structure of $G$.
Reviewer: B. A. F. Wehrfritz (London)On Brauer \(p\)-dimensions and absolute Brauer \(p\)-dimensions of Henselian fields.https://www.zbmath.org/1456.160152021-04-16T16:22:00+00:00"Chipchakov, Ivan D."https://www.zbmath.org/authors/?q=ai:chipchakov.ivan-dLet \(E\) be a field. For each prime number \(p\), let \(Brd_p(E)\) be the Brauer \(p\)-dimension of \(E\) and \(abrd_p(E)\) be the absolute Brauer \(p\)-dimension of \(E\). In this paper, the author studies the sequences \(Brd_p(E)\) and \(abrd_p(E)\), when \(p\) runs over the set \(\mathbb{P}\) of the prime numbers. Suppose that \((E,v)\) is a Henselian valued field with residue field \(\hat E\). For \(p\in\mathbb{P}\), under some restrictions on \(\hat E\), such as \(abrd_p(\hat E)=0\), the author determines
\(Brd_p(E)\) and \(abrd_p(E)\) if \(\mathrm{char}(\hat E)\not=p\). Let \(\Sigma_0\) be the set of sequences \(Brd_p(E)\) and \(abrd_p(E)\) where \(p\in \mathbb{P}\) and \(E\) runs across the class of Henselian valued fields with \(\mathrm{char}(\hat E)\not=0\) and a projective absolute Galois group \(\mathcal G_{\hat E}\). The author gives a description for \(\Sigma_0\). Especially, \(\Sigma_0\) admits a sequence \(a_p,b_p\in\mathbb{N}\cup\{0,\infty\}\), \(p\in \mathbb{P}\), with \(a_2\leq 2b_2\) and
\(a_p\geq b_p\). The paper covers similar results in the case of nonzero characteristic.
Reviewer: Ali Benhissi (Monastir)On the Hochschild homology of involutive algebras.https://www.zbmath.org/1456.160052021-04-16T16:22:00+00:00"Fernàndez-València, Ramsès"https://www.zbmath.org/authors/?q=ai:fernandez-valencia.ramses"Giansiracusa, Jeffrey"https://www.zbmath.org/authors/?q=ai:giansiracusa.jeffreyLet \(k\) be a field. An \textit{involutive vector space} is a \(k\)-vector space \(V\) equipped with an automorphism of order \(2\), denoted \(v\mapsto v^*\) and called its \textit{involution}. In other words, an involutive vector space is a \({\mathbb Z}/2\)-module. An \textit{involutive algebra} is an involutive vector space \(A\) together with an associative product such that
\[(ab)^*\,=\,b^*a^*.\]
Observe that commutative (associative) algebras are involutive algebras equipped with the identity as involution. Therefore one may think of the category of involutive algebras as some ``first-order-non-commutative-algebras'', lying in between commutative and arbitrary associative algebras.
\textit{C. Braun} defined in [J. Homotopy Relat. Struct. 9, No. 2, 317--337 (2014; Zbl 1321.16004)] a Hochschild cohomology theory for involutive algebras. Namely, let \(i\mathrm{Coder}(C)\) denote the space of involution-preserving coderivations for an involutive coalgebra \(C\). Then, as in Hochschild theory, the commutator \([m,-]\) with the product of an involutive algebra \(A\) defines a differential on the shifted tensor coalgebra on \(A\), i.e., on \(i\mathrm{Coder}(T\Sigma A)\). Braun defined the cohomology of (a complex isomorphic to) \(\Sigma^{-1}i\mathrm{Coder}(T\Sigma A)\) with respect to this differential as the Hochschild cohomology of the involutive algebra \(A\).
The authors of the article under review develop the necessary homological algebra to interpret Braun's Hochschild cohomology of an involutive algebra as an Ext functor in a suitable category. They start with a careful analysis of the category of involutive bimodules over an involutive algebra \(A\), showing that it is equivalent to the category of left modules over the \textit{involutive enveloping algebra} \(A^{ie}:=(A\otimes A^{\mathrm{op}})\otimes k[{\mathbb Z}/2]\). Then they study the Hom-Tensor adjunction as well as the correct analogues of the center and the abelianization construction of a bimodule in the involutive world.
Next follows the study of flat and projective involutive bimodules, in order to develop an involutive bar resolution. With these preliminaries in place, they pass to the Ext and Tor functors in the category of involutive bimodules. The main result is that Braun's definition and the derived functor definition compute the same cohomology, provided that the involutive algebra is projective as an involutive vector space.\par
Notably, the authors do not restrict to characteristic different from \(2\).
Reviewer: Friedrich Wagemann (Nantes)Snake graphs from triangulated orbifolds.https://www.zbmath.org/1456.051782021-04-16T16:22:00+00:00"Banaian, Esther"https://www.zbmath.org/authors/?q=ai:banaian.esther"Kelley, Elizabeth"https://www.zbmath.org/authors/?q=ai:kelley.elizabethSummary: We give an explicit combinatorial formula for the Laurent expansion of any arc or closed curve on an unpunctured triangulated orbifold. We do this by extending the snake graph construction of \textit{G. Musiker} et al. [Adv. Math. 227, No. 6, 2241--2308 (2011; Zbl 1331.13017)] to unpunctured orbifolds. In the case of an ordinary arc, this gives a combinatorial proof of positivity to the generalized cluster algebra from this orbifold.Analogues of centralizer subalgebras for fiat 2-categories and their 2-representations.https://www.zbmath.org/1456.180162021-04-16T16:22:00+00:00"Mackaay, Marco"https://www.zbmath.org/authors/?q=ai:mackaay.marco"Mazorchuk, Volodymyr"https://www.zbmath.org/authors/?q=ai:mazorchuk.volodymyr"Miemietz, Vanessa"https://www.zbmath.org/authors/?q=ai:miemietz.vanessa"Zhang, Xiaoting"https://www.zbmath.org/authors/?q=ai:zhang.xiaotingFinitary 2-categories are higher representation-theoretic analogues of finite-dimensional algebras, and the basic classification problem in higher representation theory is that of simple transitive 2-representations of a given 2-category \(\mathcal{C}\). That is, it turns out that simple transitive 2-representations are exhausted by the class of cell 2-representations, and a certain subquotient of the 2-category of Soergel bimodules over the coinvariant algebra, of type \(B_2\), is a non-elementary example. It was then applied to study simple transitive 2-representations for all small quotients of Soergel bimodules associated to finite Coxeter systems.
The main result of this manuscript says for a fiat 2-category \(\mathcal{C}\) and its 2-subcategory \(\mathcal{A}\), there is a bijection between certain classes of simple transitive 2-representations of \(\mathcal{C}\) and \(\mathcal{A}\). This reduces the problem of classification of simple transitive 2-representations for fiat 2-categories to that for fiat 2-categories with only one non-identity left, right, and two-sided cell. As an application, the authors Mackaay, Mazorchuk, Miemietz, and Zhang classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types \(B_3\) and \(B_4\).
Reviewer: Mee Seong Im (West Point)Constructing a quantum Lax pair from Yang-Baxter equations.https://www.zbmath.org/1456.812432021-04-16T16:22:00+00:00"Lima-Santos, A."https://www.zbmath.org/authors/?q=ai:lima-santos.antonioSpecht property for some varieties of Jordan algebras of almost polynomial growth.https://www.zbmath.org/1456.170172021-04-16T16:22:00+00:00"Centrone, Lucio"https://www.zbmath.org/authors/?q=ai:centrone.lucio"Martino, Fabrizio"https://www.zbmath.org/authors/?q=ai:martino.fabrizio"da Silva Souza, Manuela"https://www.zbmath.org/authors/?q=ai:souza.manuela-da-silvaLet \(F\) be a field of characteristic 0, and let \(A=UT_2(F)\) stand for the associative algebra of \(2\times 2\) upper triangular matrices. One obtains on \(A\) the structure of a Jordan algebra \(UJ_2\) by means of the Jordan product \(a\circ b=(ab+ba)/2\) for every \(a\), \(b\in A\). The Jordan algebra \(UJ_2\) admits several gradings by the cyclic group \(C_2\) of order 2. There are, up to isomorphism, three non-isomorphic non-trivial gradings. If one adds the trivial grading, and another grading by \(C_2\times C_2\), one gets the list of all gradings on \(UJ_2\). These gradings and the corresponding graded identities were described in [\textit{P. Koshlukov} and \textit{F. Martino}, J. Pure Appl. Algebra 216, No. 11, 2524--2532 (2012; Zbl 1287.17053)]. The main result of the paper under review is that in each of these five cases, the ideal of the (graded) identities of \(UJ_2\) satisfies the Specht property. This means that every ideal of (graded) identities containing the one of \(UJ_2\), is finitely generated as an ideal of (graded) identities. Additionally the authors prove that a metabelian Jordan algebra also satisfies the Specht property.
Reviewer: Plamen Koshlukov (Campinas)Study of the algebra of smooth integro-differential operators with applications.https://www.zbmath.org/1456.160212021-04-16T16:22:00+00:00"Haghany, A."https://www.zbmath.org/authors/?q=ai:haghany.ahmad"Kassaian, Adel"https://www.zbmath.org/authors/?q=ai:kassaian.adelHom-Lie-Hopf algebras.https://www.zbmath.org/1456.160332021-04-16T16:22:00+00:00"Halıcı, S."https://www.zbmath.org/authors/?q=ai:halici.serpil|halici.serspil"Karataş, A."https://www.zbmath.org/authors/?q=ai:karatas.adnan"Sütlü, S."https://www.zbmath.org/authors/?q=ai:sutlu.serkan|sutlu.s-selcukSummary: We studied both the double cross product and the bicrossproduct constructions for the Hom-Hopf algebras of general \((\alpha, \beta)\)-type. This allows us to consider the universal enveloping Hom-Hopf algebras of Hom-Lie algebras, which are of \((\alpha, \operatorname{Id})\)-type. We show that the universal enveloping Hom-Hopf algebras of a matched pair of Hom-Lie algebras form a matched pair of Hom-Hopf algebras. We observe also that, the semi-dualization of a double cross product Hom-Hopf algebra is a bicrossproduct Hom-Hopf algebra. In particular, we apply this result to the universal enveloping Hom-Hopf algebras of a matched pair of Hom-Lie algebras to obtain Hom-Lie-Hopf algebras.Polynomials from combinatorial \(K\)-theory.https://www.zbmath.org/1456.051712021-04-16T16:22:00+00:00"Monical, Cara"https://www.zbmath.org/authors/?q=ai:monical.cara"Pechenik, Oliver"https://www.zbmath.org/authors/?q=ai:pechenik.oliver"Searles, Dominic"https://www.zbmath.org/authors/?q=ai:searles.dominicSummary: We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a \(K\)-theoretic deformation of the quasi-key basis and also a lift of the \(K\)-analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the \(K\)-analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)].
The second new basis is the kaon basis, a \(K\)-theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis.
Throughout, we explore how the relationships among these \(K\)-analogues mirror the relationships among their cohomological counterparts. We make several ``alternating sum'' conjectures that are suggestive of Euler characteristic calculations.Quantum character varieties and braided module categories.https://www.zbmath.org/1456.170102021-04-16T16:22:00+00:00"Ben-Zvi, David"https://www.zbmath.org/authors/?q=ai:ben-zvi.david"Brochier, Adrien"https://www.zbmath.org/authors/?q=ai:brochier.adrien"Jordan, David"https://www.zbmath.org/authors/?q=ai:jordan.david-andrewSummary: We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants \(\int_S\mathcal{A}\) of a surface \(S\), determined by the choice of a braided tensor category \(\mathcal{A}\), and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a \textit{braided module category} for \(\mathcal{A}\), and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called \textit{quantum moment maps}. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided \(\mathcal{A}\)-modules are objects of the torus category \(\int_{T^2}\mathcal{A}\). We initiate a theory of character sheaves for quantum groups by identifying the torus integral of \(\mathcal{A}=\mathrm{Rep}_{q}G\) with the category \(\mathcal{D}_q(G/G)\)-mod of equivariant quantum \(\mathcal{D}\)-modules. When \(G=GL_n\), we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra \(\mathbb{SH}_{q,t}\).On the Morita reduced versions of skew group algebras of path algebras.https://www.zbmath.org/1456.160232021-04-16T16:22:00+00:00"Le Meur, Patrick"https://www.zbmath.org/authors/?q=ai:le-meur.patrickSummary: Let \(R\) be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita-reduced algebra associated to \(R\). \textit{I. Reiten} and \textit{C. Riedtmann} [J. Algebra 92, 224--282 (1985; Zbl 0549.16017)] proved that there exists an idempotent \(e\) of \(R\) such that the algebra \(eRe\) is both Morita equivalent to \(R\) and isomorphic to the path algebra of some quiver, which was described by
\textit{L. Demonet} [J. Algebra 323, No. 4, 1052--1059 (2010; Zbl 1210.16017)]. This article gives explicit formulas for the decomposition of any element of \(eRe\) as a linear combination of paths in the quiver described by Demonet [loc. cit.]. This is done by expressing appropriate compositions and pairings in a suitable monoidal category, which takes into account the representation theory of the finite group.Quasi-invariants in characteristic \(p\) and twisted quasi-invariants.https://www.zbmath.org/1456.812522021-04-16T16:22:00+00:00"Ren, Michael"https://www.zbmath.org/authors/?q=ai:ren.michael-s"Xu, Xiaomeng"https://www.zbmath.org/authors/?q=ai:xu.xiaomengSummary: The spaces of quasi-invariant polynomials were introduced by \textit{O. A. Chalykh} and \textit{A. P. Veselov} [Commun. Math. Phys.126, No. 3, 597-611 (1990; Zbl 0746.47025)]. Their Hilbert series over fields of characteristic 0 were computed by \textit{M. Feigin} and \textit{A. P. Veselov} [Int. Math. Res. Not. 2002, No. 10, 521-545 (2002; Zbl 1009.20044)]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, \textit{A. Braverman, P. Etingof}, and \textit{M. Finkelberg} [preprint (2020; \url{arxiv:1611.10216})] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function.Area-dependent quantum field theory.https://www.zbmath.org/1456.814112021-04-16T16:22:00+00:00"Runkel, Ingo"https://www.zbmath.org/authors/?q=ai:runkel.ingo"Szegedy, Lóránt"https://www.zbmath.org/authors/?q=ai:szegedy.lorantSummary: Area-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real number -- interpreted as area -- which behaves additively under glueing. As opposed to topological theories, in area-dependent theories the state spaces can be infinite-dimensional. We introduce the notion of regularised Frobenius algebras in Hilbert spaces and show that area-dependent theories are in one-to-one correspondence to commutative regularised Frobenius algebras. We also provide a state sum construction for area-dependent theories. Our main example is two-dimensional Yang-Mills theory with compact gauge group, which we treat in detail.Symmetries of the simply-laced quantum connections and quantisation of quiver varieties.https://www.zbmath.org/1456.812722021-04-16T16:22:00+00:00"Rembado, Gabriele"https://www.zbmath.org/authors/?q=ai:rembado.gabrieleSummary: We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.Noncommutative fiber products and lattice models.https://www.zbmath.org/1456.580072021-04-16T16:22:00+00:00"Hartwig, Jonas T."https://www.zbmath.org/authors/?q=ai:hartwig.jonas-tThis paper studies the representation theory of certain
noncommutative singular varieties using two-dimensional
lattice models. In more detail, the first main result
of the paper describes categories of weight modules over a
noncommutative biparametric deformation \(\mathcal{A}\) of
the fiber product of two Kleinian singularities of type \(A\)
in terms of periodic higher spin vertex configurations.
The second main result provides a combinatorial classification
of simple weight \(\mathcal{A}\)-modules. Finally, the third main
result describes the center of \(\mathcal{A}\), it turns our
that in some cases the center is trivial, while in some other
cases it is isomorphic to the algebra of Laurent polynomials
in one variable.
Reviewer: Volodymyr Mazorchuk (Uppsala)Auslander-Reiten quiver and representation theories related to KLR-type Schur-Weyl duality.https://www.zbmath.org/1456.160102021-04-16T16:22:00+00:00"Oh, Se-jin"https://www.zbmath.org/authors/?q=ai:oh.se-jinSummary: We introduce new partial orders on the sequence positive roots and study the statistics of the poset by using Auslander-Reiten quivers for finite type ADE. Then we can prove that the statistics provide interesting information on the representation theories of KLR-algebras, quantum groups and quantum affine algebras including Dorey's rule, bases theory for quantum groups, and denominator formulas between fundamental representations. As applications, we prove Dorey's rule for quantum affine algebras \(U_q(E_{6,7,8}^{(1)})\) and partial information of denominator formulas for \(U_q(E_{6,7,8}^{(1)})\). We also suggest conjecture on complete denominator formulas for \(U_q(E_{6,7,8}^{(1)})\).On modules satisfying the descending chain condition on cyclic submodules.https://www.zbmath.org/1456.130162021-04-16T16:22:00+00:00"Kourki, Farid"https://www.zbmath.org/authors/?q=ai:kourki.farid"Tribak, Rachid"https://www.zbmath.org/authors/?q=ai:tribak.rachidThis paper concerns the structure of modules over a commutative ring \(R\) which satisfy various generalisations of the descending chain condition (DCC) on submodules. A module is coperfect if it satisfies DCC on cyclic submodules; locally artinian if every finitely generated submodule is artinian; and semi-artinian if every non-zero factor module contains a simple submodule. The authors show that the class of coperfect modules lies properly between the classes of locally artinian and semi-artinian modules. They find conditions on \(R\) for every semiartinian module to be coperfect, and for every coperfect module to be locally artinian.
Reviewer: Phillip Schultz (Perth)Some results for the two-sided quaternionic Gabor Fourier transform and quaternionic Gabor frame operator.https://www.zbmath.org/1456.420422021-04-16T16:22:00+00:00"Li, Jinxia"https://www.zbmath.org/authors/?q=ai:li.jinxia"He, Jianxun"https://www.zbmath.org/authors/?q=ai:he.jianxunSummary: In this paper, we first present some properties of the two-sided quaternionic Gabor Fourier transform (QGFT) on quaternion valued function space \(L^2(\mathbb{R}^2,\mathbb{H})\), such as Parseval's formula and the characterization of the range of the two-sided QGFT. Then, we give the definitions of quaternionic Wiener space and quaternionic Gabor frame operator (QGFO), which are the generalizations in the quaternionic settings. Finally, we prove Walnut's and Janssen's representation theorems and other boundedness results of the QGFO.Relative cluster tilting objects in triangulated categories.https://www.zbmath.org/1456.160112021-04-16T16:22:00+00:00"Yang, Wuzhong"https://www.zbmath.org/authors/?q=ai:yang.wuzhong"Zhu, Bin"https://www.zbmath.org/authors/?q=ai:zhu.binLet \(\mathcal{D}\) be a triangulated category.
When \(\mathcal{D}\) is \(2\)-Calabi-Yau, cluster-tilting objects play a crucial role in the categorification of cluster algebras and they correspond to the clusters.
In this case, mutations of cluster-tilting objects were defined by Buan-Marsh-Reineke-Reiten-Todorov [\textit{A. B. Buan} et al., Compos. Math. 145, No. 4, 1035--1079 (2009; Zbl 1181.18006)] and \textit{O. Iyama} and \textit{Y. Yoshino} [Invent. Math. 172, No. 1, 117--168 (2008; Zbl 1140.18007)]. The mutations correspond to the mutations of clusters in the categorification
of cluster algebras. However, the mutations of cluster-tilting objects in general triangulated categories are not always possible.
The module category of the endomorphism algebra \(\Lambda=\text{End}_{\mathcal{D}}^{\mathrm{op}}(T)\) of a cluster-tilting object \(T\)
in a triangulated category \(\mathcal{D}\) is equivalent to a quotient category of this triangulated category.
Under this equivalence, \textit{D. Smith} [Ill. J. Math. 52, No. 4, 1223--1247 (2008; Zbl 1204.16009)] and \textit{C. Fu} and \textit{P. Liu} [Commun. Algebra 37, No. 7, 2410--2418 (2009; Zbl 1175.18004)] proved that a tilting module over \(\Lambda\) can be lifted
to a cluster-tilting object in \(\mathcal{D}\) and Adachi-Iyama-Reiten [\textit{T. Adachi} et al., Compos. Math. 150, No. 3, 415--452 (2014; Zbl 1330.16004)] gave a bijection between cluster-tilting objects in \(\mathcal{D}\)
and support \(\tau\)-tilting modules over \(\Lambda\), when \(\mathcal{D}\) is \(2\)-Calabi-Yau.
Unfortunately, these results do not hold if \(\mathcal{D}\) is not \(2\)-Calabi-Yau.
In this paper, the authors introduce the
notions of relative cluster tilting objects and \(T[1]\)-cluster tilting objects in a triangulated
category \(\mathcal{D}\), which are generalizations of cluster-tilting objects. Then the authors give a generalization of the
result of Adachi-Iyama-Reiten [loc. cit.]. Let \(\mathcal{D}\) be a
triangulated category with a Serre functor and a cluster-tilting object \(T\), and let
\(\Lambda=\mathrm{End}_{\mathcal{D}}^{\mathrm{op}}(T)\). Then there is an order-preserving bijection between the set of isomorphism classes of basic \(T[1]\)-cluster tilting objects in \(\mathcal{D}\) and the set of isomorphism classes of basic support \(\tau\)-tilting \(\Lambda\)-modules.
Furthermore, the authors introduce mutations of relative cluster tilting objects and give a generalization of the
result of Buan-Marsh-Reineke-Reiten-Todorov [loc. cit.] and Iyama-Yoshino [loc. cit.].
Let \(\mathcal{D}\) be a triangulated
category with a Serre functor and a cluster-tilting object \(T\). Then any basic
almost \(T[1]\)-cluster tilting object in \(\mathcal{D}\) has exactly two non-isomorphic indecomposable
complements, and they are related by exchange triangles.
As an application, the authors give a partial answer to a question of Adachi-Iyama-Reiten on exchange sequences.
Reviewer: Minghui Zhao (Beijing)Lie triple derivations of incidence algebras.https://www.zbmath.org/1456.160402021-04-16T16:22:00+00:00"Wang, Danni"https://www.zbmath.org/authors/?q=ai:wang.danni"Xiao, Zhankui"https://www.zbmath.org/authors/?q=ai:xiao.zhankuiLet \(A\) be an associative algebra over \(\mathcal{R}\), a commutative ring with unit, and let \(Z(A)\) denote the center of \(A\). An \(\mathcal{R}\)-linear map \(L\colon A\rightarrow A\) is called a Lie triple derivation if \(L([[x,y],z])=[[L(x),y],z]+[[x,L(y)],z]+[[x,y],L(z)]\) for all \(x,y,z\in A\) and where \([x,y]\) denotes the commutator of \(x\), \(y\). A Lie triple derivation is proper if it is of the form \(D+F\), where \(D\colon A\rightarrow A\) is a derivation, and \(F\colon A\rightarrow Z(A)\) is an \(\mathcal{R}\)-linear map.
From now on, \(\mathcal{R}\) denotes a \(2\)-torsion free commutative ring with unit, \(X\) denotes a locally finite preordered set, and \(I(X,\mathcal{R})\) denotes the incidence algebra of \(X\) over \(\mathcal{R}\). In the paper under review, the authors prove that if \(X\) consists of a finite number of connected components, then every Lie triple derivation of \(I(X,\mathcal{R})\) is proper.
Reviewer: Małgorzata E. Hryniewicka (Białystok)The Hopf monoid of orbit polytopes.https://www.zbmath.org/1456.051722021-04-16T16:22:00+00:00"Supina, Mariel"https://www.zbmath.org/authors/?q=ai:supina.marielSummary: Many families of combinatorial objects have a Hopf monoid structure. \textit {M. Aguiar} and \textit{F. Ardila} [``Hopf monoids and generalized permutahedra'' (2017), Preprint, \url{arXiv:1709.07504}] introduced the Hopf monoid of generalized permutahedra and showed that it contains various other notable combinatorial families as Hopf submonoids, including graphs, posets, and matroids. We introduce the Hopf monoid of orbit polytopes, which is generated by the generalized permutahedra that are invariant under the action of the symmetric group. We show that modulo normal equivalence, these polytopes are in bijection with integer compositions. We interpret the Hopf structure through this lens, and we show that applying the first Fock functor to this Hopf monoid gives a Hopf algebra of compositions. We describe the character group of the Hopf monoid of orbit polytopes in terms of noncommutative symmetric functions, and we give a combinatorial interpretation of the basic character and its polynomial invariant.On outer \((\sigma,\tau)\)-\(n\)-derivations and commutativity in prime near-rings.https://www.zbmath.org/1456.160422021-04-16T16:22:00+00:00"Aroonruviwat, Pitipong"https://www.zbmath.org/authors/?q=ai:aroonruviwat.pitipong"Leerawat, Utsanee"https://www.zbmath.org/authors/?q=ai:leerawat.utsaneeSummary: In this paper, we introduce the notion of outer \((\sigma,\tau)\)-\(n\)-derivations in a near-ring and investigate some properties. Moreover, we obtain additive commutative of a prime near-rings satisfying certain algebraic identities involving outer \((\sigma,\tau)\)-\(n\)-derivation. Furthermore, we investigate some conditions involving outer \((\sigma,\tau)\)-\(n\)-derivations for a near-ring to be a commutative ring.On finite-dimensional copointed Hopf algebras over dihedral groups.https://www.zbmath.org/1456.160312021-04-16T16:22:00+00:00"Fantino, Fernando"https://www.zbmath.org/authors/?q=ai:fantino.fernando"García, Gastón Andrés"https://www.zbmath.org/authors/?q=ai:garcia.gaston-andres"Mastnak, Mitja"https://www.zbmath.org/authors/?q=ai:mastnak.mitjaLet \(k\) be an algebraically closed field of characteristic zero, and
denote by \(\mathbb{D}_m\) the dihedral group of order \(2m\) for any
\(m\geq 3\). The aim of the paper is to classify the finite
dimensional Hopf algebras with coradical \((k\mathbb{D}_m)^*\), the
dual of the group Hopf algebra of \(\mathbb{D}_m\), for \(m=4a\geq 12\).
The approach uses the lifting method, which is very efficient for
classifying Hopf algebras with coradical a Hopf subalgebra,
deformation theory and some cohomological methods for classifying
the liftings.
Reviewer: Sorin Dascalescu (Bucureşti)On the \(\mathcal{U}_q[Sl(2)]\) Temperley-Lieb reflection matrices.https://www.zbmath.org/1456.822922021-04-16T16:22:00+00:00"Lima-Santos, A."https://www.zbmath.org/authors/?q=ai:lima-santos.antonioOdd supersymmetric Kronecker elliptic function and Yang-Baxter equations.https://www.zbmath.org/1456.160362021-04-16T16:22:00+00:00"Levin, A."https://www.zbmath.org/authors/?q=ai:levin.andrey-m"Olshanetsky, M."https://www.zbmath.org/authors/?q=ai:olshanetsky.mikhail-a"Zotov, A."https://www.zbmath.org/authors/?q=ai:zotov.andrei-vSummary: We introduce an odd supersymmetric version of the Kronecker elliptic function. It satisfies the genus one Fay identity and supersymmetric version of the heat equation. As an application, we construct odd supersymmetric extensions of the elliptic \(R\)-matrices, which satisfy the classical and the associative Yang-Baxter equations.
{\copyright 2020 American Institute of Physics}The applications of probability groups on Hopf algebras.https://www.zbmath.org/1456.160342021-04-16T16:22:00+00:00"Zhou, Jingheng"https://www.zbmath.org/authors/?q=ai:zhou.jingheng"Zhu, Shenglin"https://www.zbmath.org/authors/?q=ai:zhu.shenglinSummary: In this work, we use probability groups, introduced by
\textit{D. K. Harrison} in [Pac. J. Math. 80, 451--491 (1979; Zbl 0415.20022)], as a tool to study a semisimple Hopf algebra \(H\) with a commutative character ring and prove that the algebra generalized by the dual probability group is the center \(Z(H)\) of \(H\) and the product of two class sums is an integral combination up to a factor of \((H)^{-1}\) of the class sums of \(H\). We classify all the 2-integral probability groups with two or three elements.Truncation of unitary operads.https://www.zbmath.org/1456.180152021-04-16T16:22:00+00:00"Bao, Yan-Hong"https://www.zbmath.org/authors/?q=ai:bao.yanhong"Ye, Yu"https://www.zbmath.org/authors/?q=ai:ye.yu"Zhang, James J."https://www.zbmath.org/authors/?q=ai:zhang.james-yiming|zhang.james-jUnitary operads over a fixed field \(\Bbbk\) are those one-dimensional in arities zero and one; let \(Op_+\) denote the category of such.
A unitary operad \(P\) has a natural family of restriction operators \(P(n) \to P(s)\) for \(s \leq n\), obtained by composing with the \(0\)-ary operation in \(n-s\) places.
These restrictions are used to define a sequence of ideals \(\vphantom{\Upsilon}^k\Upsilon\) of \(P\) by intersecting some of their kernels.
An analogue of the Gelfand-Kirillov dimension of \(\Bbbk\)-algebras is given.
Namely, the GK dimension of a (locally finite-dimensional) operad \(P\) is defined by the formula
\[
\mathrm{GKdim}(P) = \limsup_{n\to \infty} \left( \log_n \left( \sum_{i=0}^n \dim_{\Bbbk} P(i)\right) \right).
\]
The authors also introduce and study other invariants, including the exponent, the signature, and the Hilbert series.
Many of the results are concerned with `2-unitary operads', which are related to `operads with multiplication' in the sense of Gerstenhaber-Voronov
[\textit{M. Gerstenhaber} and \textit{A. Voronov}, Int. Math. Res. Not. 1995, No. 3, 141--153 (1995; Zbl 0827.18004)].
More precisely, a \textbf{2-unitary} operad is an object of the comma category \(Mag\downarrow Op_+\) of operads under the magma operad, that is a unitary operad with a specified non-associative multiplication \(\mu\) (there are also associative and commutative variants of this notion).
If \(P\) is 2-unitary and has finite GK dimension, then the GK dimension must be an integer.
Further, this GK dimension can be characterized by the vanishing of the ideals \(\vphantom{\Upsilon}^k\Upsilon\).
The authors classify the 2-unitary operads of small GK dimension.
There is only a single 2-unitary operad of GK dimension 1, namely the commutative operad \(Com\).
Every augmented \(\Bbbk\)-algebra \(\Lambda\) generates a 2-unitary operad which is \(\Lambda\) in arity one and which has dimension greater than one in all higher arities so long as \(\Lambda\) is different from \(\Bbbk\).
This construction is part of an equivalence of categories between 2-unitary operads with GK dimension at most 2 and finite-dimensional, augmented \(\Bbbk\)-algebras.
The quotient operads of the associative operad \(Ass\) with fixed GK dimension at most 4 are classified when \(\Bbbk\) has characteristic zero.
There are only three such quotient operads, namely \(Ass / \vphantom{\Upsilon}^k\Upsilon\) for \(k=1,3,4\) of GK dimension \(k\).
In particular, there is no quotient operad of \(Ass\) with GK dimension 2.
It is shown that the situation is more complicated for \(k=5\).
There are several other interesting directions in the paper as well, including characterizations of artinian, semiprime operads (in the reduced, unitary, or 2-unitary cases), the fact that every signature can be realized by an object of \(Com\downarrow Op_+\), and the relationship between the GK dimension of operads and of their algebras.
Reviewer: Philip Hackney (Lafayette)Rings in which every left zero-divisor is also a right zero-divisor and conversely.https://www.zbmath.org/1456.160392021-04-16T16:22:00+00:00"Ghashghaei, E."https://www.zbmath.org/authors/?q=ai:ghashghaei.ebrahim"Koşan, M. Tamer"https://www.zbmath.org/authors/?q=ai:kosan.muhammet-tamer"Namdari, M."https://www.zbmath.org/authors/?q=ai:namdari.mehrdad"Yildirim, T."https://www.zbmath.org/authors/?q=ai:yildirim.tulayFS-coalgebras and crossed coproducts.https://www.zbmath.org/1456.160302021-04-16T16:22:00+00:00"Chen, Yuanyuan"https://www.zbmath.org/authors/?q=ai:chen.yuanyuan"Wang, Zhongwei"https://www.zbmath.org/authors/?q=ai:wang.zhongwei"Zhang, Liangyun"https://www.zbmath.org/authors/?q=ai:zhang.liangyunSummary: In this paper, we introduce FS-coalgebras, which provide solutions of FS-equations and also solution of braid equations considered by \textit{S. Caenepeel} et al. [\(K\)-Theory 19, No. 4, 365--402 (2000; Zbl 0962.16024)]. FS-coalgebras are constructed by using FS-equations and Harrison cocycles. As applications, we prove that every bialgebra \(H\) is an FS-bialgebra if and only if there is a two-sided integral \(\alpha\) in \(H^*\) such that \(\varepsilon(\alpha)=1\), and we show that the crossed coproduct \(H^R\) introduced by the Harrison cocycle \(R\) is an FS-coalgebra when \((H,R)\) is a finite-dimensional quasitriangular Hopf algebra or a Long copaired bialgebra.Some properties on the complement of the ideal-based zero divisorgraph of a near-ring.https://www.zbmath.org/1456.160432021-04-16T16:22:00+00:00"Elavarasan, B."https://www.zbmath.org/authors/?q=ai:elavarasan.balasubramanian"Porselvi, K."https://www.zbmath.org/authors/?q=ai:porselvi.kasiSummary: In this paper, we study the concepts of a \(B\)-prime ideal and a maximal \(N\)-prime ideal of \(I\) for any completely reflexive ideal \(I\) of a near-ring \(N\). We characterize the complement of the ideal-based zero-divisor graph of near-rings in terms of their connectedness and its diameter.On the combinatorics of gentle algebras.https://www.zbmath.org/1456.160072021-04-16T16:22:00+00:00"Brüstle, Thomas"https://www.zbmath.org/authors/?q=ai:brustle.thomas"Douville, Guillaume"https://www.zbmath.org/authors/?q=ai:douville.guillaume"Mousavand, Kaveh"https://www.zbmath.org/authors/?q=ai:mousavand.kaveh"Thomas, Hugh"https://www.zbmath.org/authors/?q=ai:thomas.hugh-ross"Yıldırım, Emine"https://www.zbmath.org/authors/?q=ai:yildirim.emineSummary: For \(A\) a gentle algebra, and \(X\) and \(Y\) string modules, we construct a combinatorial basis for \(\text{Hom}(X, \tau Y)\). We use this to describe support \(\tau \)-tilting modules for \(A\). We give a combinatorial realization of maps in both directions realizing the bijection between support \(\tau \)-tilting modules and functorially finite torsion classes. We give an explicit basis of \(\text{Ext}^1(Y,X)\) as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by
\textit{T. McConville} [J. Comb. Theory, Ser. A 148, 27--56 (2017; Zbl 1355.05276)], showing that many but not all of them can be extended to general gentle algebras.Free group algebras in division rings with valuation II.https://www.zbmath.org/1456.160142021-04-16T16:22:00+00:00"Sánchez, Javier"https://www.zbmath.org/authors/?q=ai:sanchez.javierSummary: We apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.
If \(L\) is a Lie algebra, we denote by \(U(L)\) its universal enveloping algebra. \textit{P. M. Cohn} [Skew fields. Theory of general division rings. Cambridge: Cambridge Univ. Press (1995; Zbl 0840.16001)] constructed a division ring \(\mathfrak{D}_L\) that contains \(U(L)\). We denote by \(\mathfrak{D}(L)\) the division subring of \(\mathfrak{D}_L\) generated by \(U(L)\).
Let \(k\) be a field of characteristic zero, and let \(L\) be a nonabelian Lie \(k\)-algebra. If either \(L\) is residually nilpotent or \(U(L)\) is an Ore domain, we show that \(\mathfrak{D}(L)\) contains (noncommutative) free group algebras. In those same cases, if \(L\) is equipped with an involution, we are able to prove that the free group algebra in \(\mathfrak{D}(L)\) can be chosen generated by symmetric elements in most cases.
Let \(G\) be a nonabelian residually torsion-free nilpotent group, and let \(k(G)\) be the division subring of the Malcev-Neumann series ring generated by the group algebra \(k[G]\). If \(G\) is equipped with an involution, we show that \(k(G)\) contains a (noncommutative) free group algebra generated by symmetric elements.
For Part I, see the author, J. Algebra 531, 221--248 (2019; Zbl 1425.16011)].Vertex operators, solvable lattice models and metaplectic Whittaker functions.https://www.zbmath.org/1456.820972021-04-16T16:22:00+00:00"Brubaker, Ben"https://www.zbmath.org/authors/?q=ai:brubaker.ben"Buciumas, Valentin"https://www.zbmath.org/authors/?q=ai:buciumas.valentin"Bump, Daniel"https://www.zbmath.org/authors/?q=ai:bump.daniel"Gustafsson, Henrik P. A."https://www.zbmath.org/authors/?q=ai:gustafsson.henrik-p-aThis paper discusses two mechanisms by which the quantum groups \(U_q (\hat{\mathfrak{g}})\), for a simple Lie algebra or superalgebra \(\mathfrak{g}\), produce families of special functions with a number of interesting properties related to functional equations, branching rules and unexpected algebraic relations. The first mechanism uses solvable lattice models associated to finite-dimensional modules of \(U_q (\hat{\mathfrak{g}})\). The second mechanism uses actions of Heisenberg and Clifford algebras on a fermionic Fock space, exploiting the boson-fermion correspondence arising in connection with soliton theory, dating back to [\textit{M. Jimbo} and \textit{T. Miwa}, Publ. Res. Inst. Math. Sci. 19, 943--1001 (1983; Zbl 0557.35091)] and pushed forward by \textit{T. Lam} [Math. Res. Lett. 13, No. 2--3, 377--392 (2006; Zbl 1160.05056)] and especially by [\textit{M. Kashiwara} et al., Sel. Math., New Ser. 1, No. 4, 787--805 (1995; Zbl 0857.17013)]. These two points of view provide new insight into the theory of metaplectic Whittaker functions for the general linear group and relate them to LLT polynomials (known also as ribbon symmetric functions). The main theorem of the paper considers two solvable lattice models, named Gamma ice and Delta, and details in Section 4 their row transfer matrices. In this study, metaplectic ice models are exploited, whose partition functions are metaplectic Whittaker functions. In the process, the authors introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to Whittaker functions. It is explained that half vertex operators agree with Lam's construction, and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials [\textit{A. Lascoux} et al., J. Math. Phys. 38, No. 2, 1041--1068 (1997; Zbl 0869.05068)] can be related to vertex operators on the quantum Fock space, only metaplectic symmetric functions are connected to solvable lattice models. A number of links with the existing literature is identified as well.
Reviewer: Piotr Garbaczewski (Opole)Matrix method for persistence modules on commutative ladders of finite type.https://www.zbmath.org/1456.550042021-04-16T16:22:00+00:00"Asashiba, Hideto"https://www.zbmath.org/authors/?q=ai:asashiba.hideto"Escolar, Emerson G."https://www.zbmath.org/authors/?q=ai:escolar.emerson-g"Hiraoka, Yasuaki"https://www.zbmath.org/authors/?q=ai:hiraoka.yasuaki"Takeuchi, Hiroshi"https://www.zbmath.org/authors/?q=ai:takeuchi.hiroshiA persistence module \(M\) on a commutative ladder over the field \(K\) can be identified with a commutative diagram of \(K\)-vector spaces and \(K\)-linear maps of the form
\[\begin{array}{cccccccc}
W_1 & \xleftrightarrow{\phi_1} & W_2 & \xleftrightarrow{\phi_2} & \cdots
& W_{n-1} & \xleftrightarrow{\phi_{n-1}} & W_n\\
\hspace{6pt}\uparrow{F_1} & & \hspace{6pt}\uparrow{F_2} &
& & \hspace{6pt}\uparrow{F_{n-1}} & & \hspace{6pt}\uparrow{F_n}\\
V_1 & \xleftrightarrow{\psi_1} & V_2 & \xleftrightarrow{\psi_2} & \cdots
& V_{n-1} & \xleftrightarrow{\psi_{n-1}} & V_n
\end{array}\]
where each horizontal arrow has a specified direction, namely either \(\leftarrow\) or \(\rightarrow\). The direction \(\tau_i\) of the arrow \(\phi_i\) is required to be the same as that of \(\psi_i\), and the sequence \(\tau=(\tau_1,\ldots,\tau_{n-1})\) is called the orientation of the ladder. In the language of zigzag persistent homology, each horizontal row in the diagram is a \(\tau\)-module [\textit{G. Carlsson} and \textit{V. de Silva}, Found. Comput. Math. 10, No. 4, 367--405 (2010; Zbl 1204.68242)]. Any such module can be written as the direct sum of interval \(\tau\)-modules of the form \({\mathbb I}[a,b]\), where \({\mathbb I}[a,b]_i=K\) if \(a\leq i\leq b\) and trivial otherwise, and all non-trivial arrows are the identity. Using this basis, the map from the bottom row of the diagram to the top row determines a matrix \(\Phi(M)\). Under the assumption that \(n\leq 4\), the authors provide a Smith normal form style algorithm for reducing \(\Phi(M)\). The persistence diagram of \(M\) can then be extracted from the resulting matrix.
Reviewer: Jason Hanson (Redmond)Matrix rings as one sided \(\sigma \)-\((S, 1)\) rings.https://www.zbmath.org/1456.160242021-04-16T16:22:00+00:00"Bhat, V. K."https://www.zbmath.org/authors/?q=ai:bhat.vijay-kumar"Singh, Pradeep"https://www.zbmath.org/authors/?q=ai:singh.pradeep-kumarLie groups of controlled characters of combinatorial Hopf algebras.https://www.zbmath.org/1456.220072021-04-16T16:22:00+00:00"Dahmen, Rafael"https://www.zbmath.org/authors/?q=ai:dahmen.rafael"Schmeding, Alexander"https://www.zbmath.org/authors/?q=ai:schmeding.alexanderA theory of controlled characters of a combinatorial Hopf algebras is introduced, given subgroups of the groups of characters. The model is
the tame Butcher group, seen as a subgroup of the Butcher-Connes-Kreimer group.
A combinatorial Hopf algebra is here a graded connected Hopf algebra, isomorphic to a polynomial algebra, with a particular basis, and the characters take their value in a fixed Banach algebra. A controlled character satisfies a growth condition given by a particular bound. If this bound is compatible with the combinatorial structure of the Hopf algebra, then the set of controlled characters is a subgroup of the group of all characters. It is proved that the group of controlled characters is an infinite-dimensional Lie group and that the underlying group is the Lie algebra of infinitesimal controlled characters. When the Hopf algebra is right-handed, it is shown that the group of controlled characters is regular in Milnor's sense.
Reviewer: Loïc Foissy (Calais)Discrete noncommutative Gel'fand Naĭmark duality.https://www.zbmath.org/1456.460612021-04-16T16:22:00+00:00"Bertozzini, Paolo"https://www.zbmath.org/authors/?q=ai:bertozzini.paolo"Conti, Roberto"https://www.zbmath.org/authors/?q=ai:conti.roberto.1"Pitiwan, Natee"https://www.zbmath.org/authors/?q=ai:pitiwan.nateeSummary: We present, in a simplified setting, a non-commutative version of the well-known Gel'fand-Naĭmark duality (between the categories of compact Hausdorff topological spaces and commutative unital \(C^*\)-algebras), where ``geometric spectra'' consist of suitable finite bundles of one-dimensional \(C^*\)-categories equipped with a transition amplitude structure satisfying saturation conditions. Although this discrete duality actually describes the trivial case of finite-dimensional \(C^*\)-algebras, the structures are here developed at a level of generality adequate for the formulation of a general topological/uniform Gel'fand-Naĭmark duality, fully addressed in a companion work.Noncommutative fibrations.https://www.zbmath.org/1456.160062021-04-16T16:22:00+00:00"Kaygun, Atabey"https://www.zbmath.org/authors/?q=ai:kaygun.atabeyThis paper shows that flat smooth extensions of associative unital algebras are reduced flat.
First, the author recalls the relevant results on reduced flat and smooth extensions. Then the reduced flat extensions and smooth extensions are identified in terms of homological conditions on the induction and restriction functors.
Then, the author shows that for reduced flat Galois fibration, the required long exact sequences in Hochschild homology and cyclic cohomology are obtained. The results are then applied to graph extension algebras.
Reviewer: Angela Gammella-Mathieu (Metz)Two-sided annihilator condition with generalized derivations on multilinear polynomials.https://www.zbmath.org/1456.160162021-04-16T16:22:00+00:00"De Filippis, V."https://www.zbmath.org/authors/?q=ai:de-filippis.vincenzo"Scudo, G."https://www.zbmath.org/authors/?q=ai:scudo.giovanni"Sorrenti, L."https://www.zbmath.org/authors/?q=ai:sorrenti.loredanaSummary: Let \(R\) be a prime ring of characteristic different from 2, with extended centroid \(C\), \(U\) its two-sided Utumi quotient ring, \(F\) a nonzero generalized derivation of \(R\), \(f(x_1, \ldots, x_n)\) a noncentral multilinear polynomial over \(C\) in \(n\) noncommuting variables, and \(a, b \in R\) such that
\[
a [F(f(r_1, \ldots, r_n)), f(r_1, \ldots, r_n)] b = 0
\]
for any \(r_1, \ldots, r_n \in R\). Then one of the following holds: (1) \(a = 0\); (2) \(b = 0\); (3) there exists \(\lambda \in C\) such that \(F(x) = \lambda x\), for all \(x \in R\); (4) there exist \(q \in U\) and \(\lambda \in C\) such that \(F(x) = (q + \lambda) x + x q\), for all \(x \in R\), and \(f(x_1, \ldots, x_n)^2\) is central valued on \(R\); (5) there exist \(q \in U\) and \(\lambda, \mu \in C\) such that \(F(x) = (q + \lambda) x + x q\), for all \(x \in R\), and \(a q = \mu a\), \(q b = \mu b\).On \(\beta\)-prime submodules.https://www.zbmath.org/1456.160042021-04-16T16:22:00+00:00"Khumprapussorn, Thawatchai"https://www.zbmath.org/authors/?q=ai:khumprapussorn.thawatchaiSummary: We introduce the concepts of \(\beta\)-prime submodules and weakly \(\beta\)-primesub modules of unital left modules over a commutative ring with nonzero identity. Some properties of these concepts are investigated. We use the notion of the product of two submodules to characterize \(\beta\)-prime submodules of a multiplication module. Characterization of \(\beta\)-prime and weakly \(\beta\)-prime submodules of arbitary modules are also given.A survey of \(s\)-unital and locally unital rings.https://www.zbmath.org/1456.160292021-04-16T16:22:00+00:00"Nystedt, Patrik"https://www.zbmath.org/authors/?q=ai:nystedt.patrikSummary: We gather some classical results and examples that show strict inclusion between the families of unital rings, rings with enough idempotents, rings with sets of local units, locally unital rings, \(s\)-unital rings and idempotent rings.Left semi-braces and solutions of the Yang-Baxter equation.https://www.zbmath.org/1456.160352021-04-16T16:22:00+00:00"Jespers, Eric"https://www.zbmath.org/authors/?q=ai:jespers.eric"van Antwerpen, Arne"https://www.zbmath.org/authors/?q=ai:van-antwerpen.arne``Braces'' are sets endowed with two operations such that they give rise to set-theoretical solutions \(r:X\times X\longrightarrow X\times X\) of the Yang-Baxter equation:
\[(r\times\operatorname{id})(\operatorname{id}\times r)(r\times\operatorname{id})=(\operatorname{id}\times r)(r\times\operatorname{id})(\operatorname{id}\times r).\]
There are different kinds of ``braces'' depending on the properties satisfied by the two operations and, as expected, these affect the associated solutions. For instance, finite involutive non-degenerate solutions correspond to ``left braces''. The authors of the present paper investigate ``braces'' in the more general sense by extending results known for some types of ``braces''.
More precisely, the authors deal with ``left semi-braces'' \((B,\cdot,\circ)\), i.e., \((B,\cdot)\) is a semigroup, \((B,\circ)\) is a group and
\[
a\circ(b\cdot c)=(a\circ b)\cdot(a\circ(\bar{a}\cdot c))
\]
for all \(a,b,c\in B\), with \(\bar{a}\) being the inverse of \(a\) in \((B,\circ)\). Under some assumptions, the authors completly describe the structure of \((B,\cdot)\) and moreover, they split \((B,\cdot,\circ)\) as a matched product of two left semi-braces with special properties. They also introduce the concept of ``ideal'' in left semi-braces. These turn out to be the kernels of the restrictions to \((B,\circ)\) of braces homomorphisms. They prove that the quotient of \(B\) by an ideal is a left semi-brace.
Finally, the authors show that left semi-braces effectively yield set-theoretical solutions of the Yang-Baxter equation and study the structure monoids associated to these solutions. Explicitly, the solution attached to \((B,\cdot,\circ)\) is the map \(r:B\times B\longrightarrow B\times B\) defined by
\[
r(a,b)=\bigl(a\circ(\bar{a}\cdot b),\overline{(\bar{a}\cdot b)}\circ b\bigr).
\]
Regarding the structure monoid, which is
\[
M(B)=\bigl\langle x\in B\mid xy=uv\quad\mbox{if}\quad r(x,y)=(u,v)\bigr\rangle,
\]
they demonstrate that, as an algebra over a any field, it is Noetherian, has finite Gelfand--Kirillov dimension and satifies a polynomial identity.
Reviewer: Cristian Vay (Córdoba)Modules over group rings of groups with restrictions on the system of all proper subgroups.https://www.zbmath.org/1456.200302021-04-16T16:22:00+00:00"Dashkova, Olga"https://www.zbmath.org/authors/?q=ai:dashkova.olga-yuSummary: We consider the class \(\mathfrak{M}\) of \(\mathbf{R}\)-modules where \(\mathbf{R}\) is an associative ring. Let \(A\) be a module over a group ring \(\mathbf{R} G\), \(G\) be a group and let \(\mathfrak{L}(G)\) be the set of all proper subgroups of \(G\). We suppose that if \(H \in \mathfrak{L}(G)\) then \(A/C_A(H)\) belongs to \(\mathfrak{M}\). We investigate an \(\mathbf{R} G\)-module \(A\) such that \(G \neq G'\), \(C_G(A) = 1\). We study the cases: 1) \(\mathfrak{M}\) is the class of all artinian \(\mathbf{R}\)-modules, \(\mathbf{R}\) is either the ring of integers or the ring of \(p\)-adic integers; 2) \(\mathfrak{M}\) is the class of all finite \(\mathbf{R}\)-modules, \(\mathbf{R}\) is an associative ring; 3) \(\mathfrak{M}\) is the class of all finite \(\mathbf{R}\)-modules, \(\mathbf{R}=F\) is a finite field.General scalar products in the arbitrary six-vertex model.https://www.zbmath.org/1456.823162021-04-16T16:22:00+00:00"Ribeiro, G. A. P."https://www.zbmath.org/authors/?q=ai:ribeiro.g-a-pColor Lie rings and PBW deformations of skew group algebras.https://www.zbmath.org/1456.170162021-04-16T16:22:00+00:00"Fryer, S."https://www.zbmath.org/authors/?q=ai:fryer.sian"Kanstrup, T."https://www.zbmath.org/authors/?q=ai:kanstrup.tina"Kirkman, E."https://www.zbmath.org/authors/?q=ai:kirkman.ellen-e"Shepler, A. V."https://www.zbmath.org/authors/?q=ai:shepler.anne-v"Witherspoon, S."https://www.zbmath.org/authors/?q=ai:witherspoon.sarah-jThe authors study color Lie rings over finite group algebras and the corresponding universal enveloping algebras [\textit{M. Scheunert}, J. Math. Phys. 20, 712--720 (1979; Zbl 0423.17003)]. More precisely, they consider such rings that arise from finite abelian groups acting diagonally on a finite dimensional vector space over a field of characteristic 0 and prove that their universal enveloping algebras can be presented as quantum Drinfeld orbifold algebras [\textit{P. Shroff}, Commun. Algebra 43, No. 4, 1563--1570 (2015; Zbl 1332.16020)]. The proof is mainly based on the theory of PBW deformations and related tools (see, e.g. [\textit{P. Shroff} and \textit{S. Witherspoon}, J. Algebra Appl. 15, No. 3, Article ID 1650049, 15 p. (2016; Zbl 1345.16025)]. As an application they show that these algebras are braided Hopf algebras.
Reviewer: Aleksandr G. Aleksandrov (Moskva)The maximum dimension of a Lie nilpotent subalgebra of \(\mathbb{M}_n(F)\) of index \(m\).https://www.zbmath.org/1456.160272021-04-16T16:22:00+00:00"Szigeti, J."https://www.zbmath.org/authors/?q=ai:szigeti.jeno"Den Berg, J. van"https://www.zbmath.org/authors/?q=ai:van-den-berg.rob"van Wyk, L."https://www.zbmath.org/authors/?q=ai:van-wyk.leon"Ziembowski, M."https://www.zbmath.org/authors/?q=ai:ziembowski.michalThis paper under review is an attempt to answer a conjecture posed by \textit{J. Szigeti} and \textit{L. Van Wyk} in [Commun. Algebra 43, No. 11, 4783--4796 (2015; Zbl 1333.16003)]. The statement of this conjecture is rendered less cumbersome if expressed in terms of a function \(M(\ell, n)\) of positive integer arguments \(\ell\) and \(n\), defined as follows:
\[
\begin{aligned}
M(\ell, n) = & \max \Bigg\{\, \frac{1}{2} \left(n^2- \sum \limits _{i=1}^{\ell} k_i^2\right)+1 \, | \, k_1, k_2, \cdots, k_{\ell}\,\\
&
\text{are nonnegative integers such that}\, \sum_{i=1}^{\ell} k_i=n \, \Bigg\}
\end{aligned}
\]
Szigeti and van Wyk's conjecture is the following: if \(F\) is any field and \(R\) any \(F\)-subalgebra of the algebra
\(\mathbb{M}_n(F)\) of \(n\times n\) matrices over \(F\) with Lie nilpotence index \(m\), then
\[
\dim_F R\leq M(m+1, n).
\]
This conjecture is eventually solved in the current joint work by the four authors. The case \(m=1\) reduces to a classical theorem of \textit{I. Schur} [J. Reine Angew. Math. 130, 66--76 (1905; JFM 36.0140.01)], later generalized by \textit{N. Jacobson} [Bull. Am. Math. Soc. 50, 431--436 (1944; Zbl 0063.03016)] to all fields, which asserts that if \(F\) is an algebraically closed field of characteristic zero and \(R\) is any commutative F-subalgebra of \(\mathbb{M}_n(F)\), then \(\dim_FR\leq \lfloor \frac{n^2}{4} \rfloor+1\). Examples constructed from block upper triangular matrices show that the upper bound of \(M(m+1, n)\) cannot be lowered for any choice of \(m\) and \(n\). An explicit formula for \(M(m+1, n)\) is also derived simultaneously.
Reviewer: Wei Feng (Beijing)LCD codes and self-orthogonal codes in generalized dihedral group algebras.https://www.zbmath.org/1456.140332021-04-16T16:22:00+00:00"Gao, Yanyan"https://www.zbmath.org/authors/?q=ai:gao.yanyan"Yue, Qin"https://www.zbmath.org/authors/?q=ai:yue.qin"Wu, Yansheng"https://www.zbmath.org/authors/?q=ai:wu.yanshengA group code is a right ideal in a group ring \(R[G]\) where \(R\) is a commutative ring and \(G\) a finite group. In this paper, the authors consider a finite field \(\mathbb{F}_q\) and a generalized dihedral group \(D_{2n,r}\), \(\gcd(2n,q)=1\). As a first main result, they explicitly describe the primitive idempotents of the group algebra \(\mathbb{F}_q[D_{2n,r}]\). Given a code \(C\), it is named LCD whenever \(C \cap C^\perp = \{0\}\) and it is self-orthogonal iff \(C \subseteq C^\perp\), where \(C^\perp\) means dual of \(C\). LCD codes have cryptographic applications and self-orthogonal codes provide quantum codes. The second main result of the paper describes and counts LCD and self-orthogonal group codes in \(\mathbb{F}_q[D_{2n,r}]\).
Reviewer: Carlos Galindo (Castellón)On finite strongly critical rings.https://www.zbmath.org/1456.160202021-04-16T16:22:00+00:00"Mal'tsev, Yuriĭ Nikolaevich"https://www.zbmath.org/authors/?q=ai:maltsev.yurii-nikolaevich"Zhuravlev, Evgeniĭ Vladimirovich"https://www.zbmath.org/authors/?q=ai:zhuravlev.evgenii-vladimirovichThe authors consider properties of finite (associative) rings. Recall that a finite ring \(A\) is critical if \(A\) does not lie in the variety generated by all of its proper homomorphic images. It is well known that a critical finite ring is a subdirectly irreducible, and its order is \(p^n\) for some prime \(p\). A ring \(R\) of order \(n\) is strongly critical if it does not belong to the variety generated by all rings of order less than \(n\). The authors prove the following results. If \(R\) is a simple finite ring then it is strongly critical. The same conclusion holds if \(R\) is critical and of order \(p^2\) where \(p\) is prime. They give an example of a ring of order 8 which is critical but not strongly critical. Moreover, if \(R\) is a finite ring and \(M_n(R)\) is strongly critical then \(R\) itself is strongly critical. The converse also holds. Finally if \(R\) is finite, and \(R/J(R)\cong M_n(GF(q))\) , and if \(J(R)\) is strongly critical then \(R\) itself is strongly critical. Here \(J(R)\) is the radical of \(R\), and \(GF(q)\) is the field with \(q\) elements.
Reviewer: Plamen Koshlukov (Campinas)50 years of finite geometry, the ``geometries over finite rings'' part.https://www.zbmath.org/1456.510022021-04-16T16:22:00+00:00"Keppens, Dirk"https://www.zbmath.org/authors/?q=ai:keppens.dirkSummary: Whereas for a substantial part, ``finite geometry'' during the past 50 years has focussed on geometries over finite fields, geometries over finite rings that are not division rings have got less attention. Nevertheless, several important classes of finite rings give rise to interesting geometries.
In this paper we bring together some results, scattered over the literature, concerning finite rings and plane projective geometry over such rings. The paper does not contain new material, but by collecting information in one place, we hope to stimulate further research in this area for at least another 50 years of finite geometry.On almost subnormal subgroups and maximal subgroups in skew linear groups.https://www.zbmath.org/1456.160122021-04-16T16:22:00+00:00"Dung, Truong Huu"https://www.zbmath.org/authors/?q=ai:dung.truong-huuIn [J. Algebra 114, No. 2, 261--267 (1988; Zbl 0645.16013)], \textit{L. Makar-Limanov} proved that if \(D\) is a division ring of infinite dimension over its center \(F\), then any subnormal subgroup of \(D^\times\) satisfying a generalized Laurent polynomial identity over \(D\) is central. The author's first aim in this paper is to generalize the above results for almost subnormal subgroups of \(\mathrm{GL}_n(D)\) satisfying a generalized Laurent polynomial identity in the case when \(D\) is algebraic and infinite dimensional over its uncountable center \(F\). Here, an almost normal subgroup \(H\) of a group \(G\) is a group for which there exists a family of subgroups \(H = H_r \leq H_{r-1} \leq \dots \leq H_1 = G\) of \(G\) such that for each \(i \in \{1,\dots,r\}\), either \(H_i\) is normal in \(H_{i-1}\) or \(H_i\) has infinite index in \(H_{i-1}\). More explicitly, the author proves that if \(N\) is an almost subnormal subgroup of \(\mathrm{GL}_n(D)\) satisfying a generalized Laurent polynomial identity, then \(N\) is central (Theorem 2.5). Then the author continues with maximal subgroups of \(\mathrm{GL}_n(D)\) satisfying a Laurent polynomial identity by proving that if \(M\) is a maximal subgroup of \(\mathrm{GL}_n(D)\) such that \(D\) is infinite dimensional over its infinite center \(F\) and \(F[M]\) is algebraic over \(F\), then \(M\) is absolutely irreducible (Theorem 2.6). The author also investigates the maximal subgroups of an almost subnormal subgroup of \(D^\times\). In [\textit{M. Ramezan-Nassab} and \textit{D. Kiani}, J. Algebra 399, 269--276 (2014; Zbl 1304.20064)], maximal subgroups of subnormal subgroups of \(D^\times\) were studied and it was shown that every nilpotent maximal subgroup of a subnormal subgroup of \(D^\times\) is abelian. The author extends this result for any maximal subgroup \(M\) of a non-central almost subnormal subgroup of \(D^\times\) in the case when \(D\) is infinite dimensional over its infinite center \(F\) and \(C_D(M)\setminus F\) contains an algebraic element over \(F\). Namely, we show that if \(M\) satisfies a Laurent polynomial identity, then \(M\) is abelian (Theorem 2.10).
Reviewer: Adam Chapman (Tel Hai)Lie (Jordan) derivations of arbitrary triangular algebras.https://www.zbmath.org/1456.160412021-04-16T16:22:00+00:00"Wang, Yu"https://www.zbmath.org/authors/?q=ai:wang.yuThe author constructs a triangular algebra from a given triangular algebra, using the notion of maximal left (right) ring of quotients (Theorem 2.1: Let $U=\text{Tri}(A, M,B)$ be a triangular algebra. Set $U^0$ as $2\times 2$ a matrix such that $a_{11}= A((\tau_r^{-1}) Z(B))$, $a_{12}=M$, $a_{21}=0$, $a_{22}=B((\tau_l )Z(A))$. Then $U^0$ is a triangular algebra such that $U$ is a subalgebra of $U^0$ having the same unity, where $\tau$ is a unique algebra isomorphism).
Also, the author has a description of Lie derivations on arbitrary triangular algebras through the constructed triangular algebra (Theorem 3.1: Let $U=\text{Tri}(A, M, B)$ be a triangular algebra. Let $L$ be a Lie derivation on $U$. Then there exists a triangular algebra $U^0$ such that $U$ is a subalgebra of $U^0$ having the same unity and $L$ can be written as $L=\delta +\tau$, where $\delta: U\to U^0$ is a derivation and $\tau: U\to Z( U^0)$ is a linear map such that $\tau ([x, y]) = 0$ for all $x, y\in U$).
Consequently, the author obtains a description of Jordan derivations on arbitrary triangular algebras (Corollary 3.1. Let $U =\text{Tri} (A, M, B)$ be a triangular algebra. Let $D$ be a Jordan derivation on $U$. Suppose that $U$ is 2-torsion free. Then $D$ is a derivation. Otherwise, there exists a triangular algebra $U^0$ such that $U$ is a subalgebra of $U^0$ having the same unity and $D$ can be written as $D=\delta+\tau$, where $\delta:U\to U^0$ is a derivation and $\tau: U\to Z (U^0)$ is a linear map such that $\tau(x\circ y) =0$ for all $x, y\in U$).
Finally, the author presents other results like propositions, lemmas and corollaries which help him to reach his aim.
Reviewer: Mehsin Atteya (Leicester)Generalized Gaudin systems in an external magnetic field and reflection equation algebras.https://www.zbmath.org/1456.823232021-04-16T16:22:00+00:00"Skrypnyk, T."https://www.zbmath.org/authors/?q=ai:skrypnyk.taras-vOn derivations involving prime ideals and commutativity in rings.https://www.zbmath.org/1456.160172021-04-16T16:22:00+00:00"Mamouni, A."https://www.zbmath.org/authors/?q=ai:mamouni.abdellah"Oukhtite, L."https://www.zbmath.org/authors/?q=ai:oukhtite.lahcen"Zerra, M."https://www.zbmath.org/authors/?q=ai:zerra.mAn additive mapping \(d\) defined on a ring \(R\) is called a derivation if \(d(xy)=d(x)y+xd(y)\) for all \(x,y \in R\). The results of this paper are separated in two parts. The first part is related to derivations involving prime ideals and the second part explains some special derivations. In the first result of the first part, the authors give some consequences for derivations \(d_1\) and \(d_2\) in case \([d_1(x),d_2(y)]\in P\) for all \(x,y \in R\), where \(P\) is a prime ideal of \(R\). Furthermore, the authors give some consequences for derivations \(d_1\) and \(d_2\) in case \(d_1(x)\circ d_2(y)\in P\) for all \(x,y \in R\), where \(P\) is a prime ideal of \(R\). Now, if the derivations \(d_1\) and \(d_2\) satisfy one of the following conditions: \([d_1(x),y]+[x,d_2(y)]\in P\) for all \(x,y \in R\), or \([d_1(x),y]+[x,d_2(y)]-[x,y]\in P\) for all \(x,y \in R\), or \([d_1(x),d_2(y)]-[d_2(y),x]-[y,d_1(x)]\in P\) for all \(x,y \in R\) then we have \(d_1(R) \subseteq P\) and \(d_2(R) \subseteq\) or \(R/P\) is a commutative integral domain. This property is explained in Theorem 3 of this paper. In the final result of this paper, the authors give some consequences if a derivation \(d\) defined on a ring \(R\) satisfies the following conditions \(d(x,y)-d(x)d(y) \in P\) for all \(x,y \in R\) (respectively, \(d(x,y)-d(y)d(x) \in P\) for all \(x,y \in R\)).
Reviewer: Puguh Wahyu Prasetyo (Yogyakarta)Duality for convex monoids.https://www.zbmath.org/1456.812682021-04-16T16:22:00+00:00"Roumen, Frank"https://www.zbmath.org/authors/?q=ai:roumen.frank"Roy, Sutanu"https://www.zbmath.org/authors/?q=ai:roy.sutanuSummary: Every \(C^*\)-algebra gives rise to an effect module and a convex space of states, which are connected via Kadison duality. We explore this duality in several examples, where the \(C^*\)-algebra is equipped with the structure of a finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra or group algebra of a finite group, the resulting state spaces form convex monoids. We will prove that both these convex monoids can be obtained from the other one by taking a coproduct of density matrices on the irreducible representations. We will also show that the same holds for a tensor product of a group and a function algebra.The Grassmann algebra over arbitrary rings and minus sign in arbitrary characteristic.https://www.zbmath.org/1456.160192021-04-16T16:22:00+00:00"Dor, Gal"https://www.zbmath.org/authors/?q=ai:dor.gal"Kanel-Belov, Alexei"https://www.zbmath.org/authors/?q=ai:kanel-belov.alexei"Vishne, Uzi"https://www.zbmath.org/authors/?q=ai:vishne.uziThe Grassmann (also known as exterior) algebra \(G\) is widely used in several areas of mathematics and also in theoretical physics. It provides the most natural and fundamental example of a superalgebra. It is usually defined as the associative algebra generated by the elements \(e_1\), \(e_2\), \dots, subject to the relations \(e_ie_j=-e_je_i\) for every \(i\) and \(j\); in particular \(e_i^2=0\). The Grassmann algebra became extremely important in PI theory. It is the easiest example of an algebra that satisfies polynomial identities (the triple commutator \([x,y,z]\)) but in characteristic 0 does not satisfy any standard identity. Later on it was an essential tool in the theory developed by \textit{A. R. Kemer} [Ideals of identities of associative algebras. Transl. from the Russian by C. W. Kohls. Transl. ed. by Ben Silver. Providence, RI: American Mathematical Society (1991; Zbl 0732.16001)]. It also appears in the definition of a superalgebra (not necessarily associative).
It is well known that the identities of \(G\) follow from the triple commutator whenever the base field is infinite and of characteristic different from 2. The problems arise in characteristic 2: the above definition of \(G\) makes it commutative. In 2000, the second-named author of the paper under review constructed an analogue \(G^+\) of the Grassmann algebra in characteristic 2, and used it to construct one of the first examples of associative algebras (in characteristic 2) whose identities do not admit any finite basis, see [\textit{A. Ya. Belov}, Sb. Math. 191, No. 3, 13--24 (2000; Zbl 0960.16029); translation from Mat. Sb. 191, No. 3, 13--24 (2000)].
Generalising the Grassmann algebra \(G\) one comes naturally to the notion of the free superalgebra \(S\), it is the tensor product of a polynomial algebra and of the Grassmann algebra. Denote the supercommutator of \(a\) and \(b\) by \(\{a,b\}\).
The authors consider algebras over a commutative and associative unital ring \(C\). They construct an algebra \(\mathfrak{G}\) over \(C\) whose behaviour is similar to that of \(G\). We recall here that the above mentioned algebra \(G^+\) can take the place of \(G\) only in characteristic 2. So the algebra \(\mathfrak{G}\) is indeed a characteristic-free generalisation of \(G\). They prove that the ideal of identities of \(\mathfrak{G}\) is generated by the triple commutator. If \(1/2\in C\) they prove that \(\mathfrak{G}\) is PI equivalent to the free supercommutative algebra \(S\) and moreover for every \(C\)-algebra \(A\) the identities of \(A\otimes_C \mathfrak{G}\) and of \(A\otimes_C S\) are the same.
If \(C\) is a field of characteristic not 2, it is well known that for every permutation \(\sigma\) of the symmetric group \(S_n\), the equality \(e_{\sigma(1)} \cdots e_{\sigma(n)} = (-1)^\sigma e_1\cdots e_n\) holds in \(G\) where \((-1)^\sigma\) is the sign of the permutation \(\sigma\). The authors define an analogue to the sign, denoted by \(\mathfrak{sgn}\), in the setting of \(\mathfrak{G}\). This generalised sign admits a cohomological interpretation exploited by the authors. They use it in order to show that the codimension sequence of \(\mathfrak{G}\) behaves exactly like the one for \(G\), namely the \(n\)-th codimension of \(\mathfrak{G}\) equals \(2^{n-1}\).
The authors also consider generalised superalgebras (called in the paper \(\Sigma\)-superalgebras) in the following sense: these are graded by an infinite but enumerable group of exponent 2 (which must be abelian). They construct the free \(\Sigma\)-supercommutative algebra \(\mathfrak{S}\). Afterwards they generalise the construction of the Grassmann hull (used first by Kemer, see the above reference), to the generalised Grassmann hull of a \(\Sigma\)-superalgebra. The authors study in detail the ideal of identities of this algebra.
\par
In the case of algebras over a field of characteristic different from 2, one defines the supertrace in a canonical way using the properties of the Grassmann algebra. The authors develop the notion of a supertrace to \(\Sigma\)-supertraces; it is characteristic-free. They define the free \(\Sigma\)-supertrace algebra and the corresponding notion of \(\Sigma\)-supertrace identities. They also develop a theory of identities with a linear function.
Reviewer: Plamen Koshlukov (Campinas)An alternative perspective on pure-projectivity of modules.https://www.zbmath.org/1456.160012021-04-16T16:22:00+00:00"Alagöz, Yusuf"https://www.zbmath.org/authors/?q=ai:alagoz.yusuf"Dur\~{g}un, Yılmaz"https://www.zbmath.org/authors/?q=ai:durgun.yilmazSummary: The study of pure-projectivity is accessed from an alternative point of view. Given modules \(M\) and \(N,M\) is said to be \(N\)-pure-subprojective if for every pure epimorphism \(g: B\rightarrow N\) and homomorphism \(f:M\rightarrow N\), there exists a homomorphism \(h:M\rightarrow B\) such that \(gh=f\). For a module \(M\), the pure-subprojectivity domain of \(M\) is defined to be the collection of all modules \(N\) such that \(M\) is \(N\)-pure-subprojective. We obtain characterizations for various types of rings and modules, including \textit{FP}-injective and \textit{FP}-projective modules, von Neumann regular rings and pure-semisimple rings in terms of pure-subprojectivity domains. As pure-subprojectivity domains clearly include all pure-projective modules, a reasonable opposite to pure-projectivity in this context is obtained by considering modules whose pure-subprojectivity domain consists of only pure-projective. We refer to these modules as \textit{psp}-poor. Properties of pure-subprojectivity domains and of \textit{psp}-poor modules are studied.Combinatorics of exceptional sequences in type A.https://www.zbmath.org/1456.160092021-04-16T16:22:00+00:00"Garver, Alexander"https://www.zbmath.org/authors/?q=ai:garver.alexander"Igusa, Kiyoshi"https://www.zbmath.org/authors/?q=ai:igusa.kiyoshi"Matherne, Jacob P."https://www.zbmath.org/authors/?q=ai:matherne.jacob-p"Ostroff, Jonah"https://www.zbmath.org/authors/?q=ai:ostroff.jonahSummary: Exceptional sequences are certain sequences of quiver representations. We introduce a class of objects called strand diagrams and use these to classify exceptional sequences of representations of a quiver whose underlying graph is a type \(\mathbb{A}_n\) Dynkin diagram. We also use variations of these objects to classify \(\mathbf c\)-matrices of such quivers, to interpret exceptional sequences as linear extensions of explicitly constructed posets, and to give a simple bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions.Coneat injective modules.https://www.zbmath.org/1456.160022021-04-16T16:22:00+00:00"Hamid, Mohanad Farhan"https://www.zbmath.org/authors/?q=ai:hamid.mohanad-farhanSummary: A module is called \textit{coneat injective} if it is injective with respect to all coneat exact sequences. The class of such modules is enveloping and falls properly between injectives and pure injectives. Generalizations of coneat injectivity, like relative coneat injectivity and full invariance of a module in its coneat injective envelope, are studied. Using properties of such classes of modules, we characterize certain types of rings like von Neumann regular and right SF-rings. For instance, \(R\) is a right SF-ring if and only if every coneat injective left \(R\)-module is injective.Some minimal rings related to 2-primal rings.https://www.zbmath.org/1456.160182021-04-16T16:22:00+00:00"Szabo, Steve"https://www.zbmath.org/authors/?q=ai:szabo.steveSummary: In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS I were given in order to show that the ring class inclusions were strict. Many of the rings given in the examples were infinite. In this paper, where possible, examples of minimal finite rings of the various types are given. Along with the rings in the previous taxonomy, NI, abelian and reflexive rings are also included.Tropical generalized interval systems.https://www.zbmath.org/1456.000522021-04-16T16:22:00+00:00"Albini, Giovanni"https://www.zbmath.org/authors/?q=ai:albini.giovanni"Bernardi, Marco Paolo"https://www.zbmath.org/authors/?q=ai:bernardi.marco-paoloSummary: This paper aims to refine the formalization of David Lewin's Generalized Interval System (GIS) by the means of tropical semirings. Such a new framework allows to broaden the GIS model introducing a new operation and consequently new musical and conceptual insights and applications, formalizing consistent relations between musical elements in an original unified structure. Some distinctive examples of extensions of well-known infinite GIS for lattices are then offered and the impossibility to build tropical GIS in the finite case is finally proven and discussed.
For the entire collection see [Zbl 1425.00082].On weakly locally finite division rings.https://www.zbmath.org/1456.160132021-04-16T16:22:00+00:00"Deo, Trinh Thanh"https://www.zbmath.org/authors/?q=ai:deo.trinh-thanh"Bien, Mai Hoang"https://www.zbmath.org/authors/?q=ai:mai-hoang-bien."Hai, Bui Xuan"https://www.zbmath.org/authors/?q=ai:hai.bui-xuanIt is first shown that (1) a division ring is weakly locally finite if and only if it is locally PI, and (2) there are no weakly locally finite division ring whose Gelfand-Kirillov dimension is non-integer. Then for each integer \(n\geq0\) or \(n=\infty\), a weakly locally finite division ring with Gelfand-Kirillov dimension \(n\) is constructed. In particular, this shows that there exist infinitely many weakly locally finite division rings that are not locally finite.
The following results related to Kurosh problems and Herstein's conjectures are proved for the class of weakly locally finite division rings: Let \(D\) be a division ring with center \(F\).
(1) \(D\) is locally finite if and only if \(D\) is weakly locally finite and algebraic.
(2) If \(D\) is non-commutative algebraic, weakly locally finite, and \(N\) a subgroup of \(\text{GL}_{n}(D)\) containing \(F^{\ast}\), \(n\geq1\), then \(N\) is not finitely generated.
(3) If \(N\) is a subnormal subgroup of \(D^{\ast}\) radical over \(F\), then \(N\) is contained in \(F\).
Reviewer: Wen-Fong Ke (Tainan)Cyclic cohomology and Chern-Connes pairing of some crossed product algebras.https://www.zbmath.org/1456.580082021-04-16T16:22:00+00:00"Quddus, Safdar"https://www.zbmath.org/authors/?q=ai:quddus.safdarSummary: We compute the cyclic and Hochschild cohomology groups for the algebras \(\mathcal{A}_\theta^{alg} \rtimes \mathbb{Z}_3\), \(\mathcal{A}_\theta^{alg} \rtimes \mathbb{Z}_4\) and \(\mathcal{A}_\theta^{alg} \rtimes \mathbb{Z}_6\). We also compute the partial Chern-Connes index table for each of these algebras.Deformation theories controlled by Hochschild cohomologies.https://www.zbmath.org/1456.160282021-04-16T16:22:00+00:00"Carolus, Samuel"https://www.zbmath.org/authors/?q=ai:carolus.samuel"Hokamp, Samuel A."https://www.zbmath.org/authors/?q=ai:hokamp.samuel-a"Laubacher, Jacob"https://www.zbmath.org/authors/?q=ai:laubacher.jacobSummary: We explore how the higher order Hochschild cohomology controls a deformation theory when the simplicial set models the 3-sphere. Besides generalizing to the \(d\)-sphere for any \(d\ge 1\), we also investigate a deformation theory corresponding to the tertiary Hochschild cohomology, which naturally reduces to those studied for the secondary and usual Hochschild cohomologies under certain conditions.On group rings and some of their applications to combinatorics and symmetric cryptography.https://www.zbmath.org/1456.940582021-04-16T16:22:00+00:00"Carlet, Claude"https://www.zbmath.org/authors/?q=ai:carlet.claude"Tan, Yin"https://www.zbmath.org/authors/?q=ai:tan.yinSummary: We give a survey of recent applications of group rings to combinatorics and to cryptography, including their use in the differential cryptanalysis of block ciphers.Sylow like theorems for \(V(\mathbb{Z}G)\).https://www.zbmath.org/1456.160372021-04-16T16:22:00+00:00"Kimmerle, Wolfgang"https://www.zbmath.org/authors/?q=ai:kimmerle.wolfgangSummary: The main part of this article is a survey on torsion subgroups of the unit group of an integral group ring. It contains the major parts of my talk given at the conference ``Groups, Group Rings and Related Topics'' at UAEU in Al Ain October 2013. In the second part special emphasis is layed on \(p\)-subgroups and on the open question whether there is a Sylow like theorem in the normalized unit group of an integral group ring. For specific classes of finite groups we prove that \(p\)-subgroups of the normalized unit group of its integral group rings \(V(\mathbb{Z}G)\) are isomorphic to subgroups of \(G \). In particular for \(p = 2\) this is shown when \(G\) has abelian Sylow 2-subgroups. This extends results known for soluble groups to classes of groups which are not contained in the class of soluble groups.The endomorphisms algebra of translations group and associative unitary ring of trace-preserving endomorphisms in affine plane.https://www.zbmath.org/1456.510012021-04-16T16:22:00+00:00"Zaka, Orgest"https://www.zbmath.org/authors/?q=ai:zaka.orgest"Mohammed, Mohanad A."https://www.zbmath.org/authors/?q=ai:mohammed.mohanad-aSummary: A description of Endomorphisms of the translation group is introduced in an affine plane, will define the addition and composition of the set of endomorphisms and specify the neutral elements associated with these two actions and present the Endomorphism algebra thereof will distinguish the Trace-preserving endomorphism algebra in affine plane, and prove that the set of Trace-preserving endomorphism associated with the `addition' action forms a commutative group. We also try to prove that the set of trace-preserving endomorphism, together with the two actions, in it, `addition' and `composition' forms an associative and unitary ring.Group rings for communications.https://www.zbmath.org/1456.941352021-04-16T16:22:00+00:00"Hurley, Ted"https://www.zbmath.org/authors/?q=ai:hurley.tedSummary: This is a survey of some recent applications of abstract algebra, and in particular group rings, to the communications' areas.Higher Auslander algebras of type \(\mathbb{A}\) and the higher Waldhausen \(\mathsf{S}\)-constructions.https://www.zbmath.org/1456.180112021-04-16T16:22:00+00:00"Jasso, Gustavo"https://www.zbmath.org/authors/?q=ai:jasso.gustavoSummary: These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with \textit{T. Dyckerhoff} et al. [Adv. Math. 355, Article ID 106762, 73 p. (2019; Zbl 07107210)]. In them we relate Iyama's higher Auslander algebras of type \(\mathbb{A}\) to Eilenberg-Mac Lane spaces in algebraic topology and to higher-dimensional versions of the Waldhausen \(\mathsf{S}\)-construction from algebraic \(K\)-theory.
For the entire collection see [Zbl 07314259].On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3.https://www.zbmath.org/1456.112242021-04-16T16:22:00+00:00"Hu, Yong"https://www.zbmath.org/authors/?q=ai:hu.yong"Wu, Zhengyao"https://www.zbmath.org/authors/?q=ai:wu.zhengyaoSummary: Let \(F\) be a field, \(\ell\) a prime and \(D\) a central division \(F\)-algebra of \(\ell\)-power degree. By the Rost kernel of \(D\) we mean the subgroup of \(F^*\) consisting of elements \(\lambda\) such that the cohomology class \((D)\cup (\lambda)\in H^3(F,\mathbb{Q}_{\ell}/\mathbb Z_{\ell}(2))\) vanishes. In ~1985, \textit{A. A. Suslin} [in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 25, 115--207 (1984; Zbl 0558.12013)] conjectured that the Rost kernel is generated by \(i\)-th powers of reduced norms from \(D^{\otimes i}\) for all \(i\ge 1\). Despite known counterexamples, we prove some new special cases of Suslin's conjecture. We assume \(F\) is a henselian discrete valuation field with residue field \(k\) of characteristic different from \(\ell\). When \(D\) has period \(\ell \), we show that Suslin's conjecture holds if either \(k\) is a \(2\)-local field or the cohomological \(\ell\)-dimension \(\operatorname{cd}_{\ell}(k)\) of \(k\) is \(\le 2\). When the period is arbitrary, we prove the same result when \(k\) itself is a henselian discrete valuation field with \(\operatorname{cd}_{\ell}(k)\le 2\). In the case \(\ell=\operatorname{char}(k)\), an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.Algebraic dynamics.https://www.zbmath.org/1456.370132021-04-16T16:22:00+00:00"Manes, Ernie"https://www.zbmath.org/authors/?q=ai:manes.ernie-gSummary: Dynamical notions are introduced in arbitrary tight categories. The enveloping semigroup of \(X\) is the free object on one generator in the variety generated by \(X\). Two new examples are dynamical systems in which all spaces are countably tight and compact spaces which are homeomorphic to their square. All dynamic varieties have a universal minimal object. Comfort types are identified with certain singly-generated submonads of the ultrafilter monad.On the heredity of \(V\)-modules over Noetherian nonsingular rings.https://www.zbmath.org/1456.160032021-04-16T16:22:00+00:00"Soontharanon, Jarunee"https://www.zbmath.org/authors/?q=ai:soontharanon.jarunee"Huynh, Dinh Van"https://www.zbmath.org/authors/?q=ai:huynh.dinh-van"Sanh, Nguyen Van"https://www.zbmath.org/authors/?q=ai:nguyen-van-sanh.Valued modules on skew polynomial rings and Bézout domains.https://www.zbmath.org/1456.130062021-04-16T16:22:00+00:00"Point, Françoise"https://www.zbmath.org/authors/?q=ai:point.francoiseSummary: We review results obtained on the model theory of valued modules over skew polynomial rings and Bézout domains in the definablility and decidability point of view.
For the entire collection see [Zbl 1419.03005].Higher derivations of finitary incidence algebras.https://www.zbmath.org/1456.160262021-04-16T16:22:00+00:00"Kaygorodov, Ivan"https://www.zbmath.org/authors/?q=ai:kaigorodov.i-b"Khrypchenko, Mykola"https://www.zbmath.org/authors/?q=ai:khrypchenko.mykola-s"Wei, Feng"https://www.zbmath.org/authors/?q=ai:wei.fengA sequence of additive maps \((d_n)_{n\in \mathbb N}\) on a unital ring \(R\) is called a higher derivation if the identities
\[ d_0(x)=x\quad \hbox{ and } \quad d_n(xy) = \sum_{k=0}^n d_k(x)d_{n-k}(y)\] hold. Examples include the sequence of additive maps \(d_n\colon x\mapsto r^{n-1}(rx-xr)\), with an element \(r\in R\) kept fixed, as well as, when \(R\) is an algebra over a field with characteristic \(0\), the sequence \((\frac{1}{n!}d^n)_{n\in\mathbb N}\) with \(d\colon R\to R\) being a usual derivation on \(R\).
In fact, higher derivations are in one-to-one, onto correspondence with those automorphisms \(\alpha\) of the ring of formal power series \(R[[t]]\) which fix an indeterminate \(t\) and map each \(x\in R\subseteq R[[t]]\) into the set \(x+t R[[t]]\); the correspondence is given by \(\alpha(x)=\sum d_{n}(x)t^n\); \(x\in R\subseteq R[[t]]\).
The main result of the paper under review describes the form of \(R\)-linear higher derivations on finitary incidence algebras \(FI(R)\) over commutative unital rings \(R\). Here, by definition, \(FI(R)\) is an \(R\)-algebra of \(R\)-valued functions with domain consisting of all pairs \((x,y)\), ordered in a given preordered set \(P\), which have a finite support when restricted to each of the subsets \(\Omega_{(x,y)}:=\{(u,v)\in P^2;\;\; x\le u<v\le y\}\). The \(R\)-module structure on \(FI(R)\) is standard and the multiplication is convolution-like
\[(f\ast g)(x,y):=\sum_{x\le z\le y} f(x,z)g(z,y).\]
Reviewer: Bojan Kuzma (Ljubljana)Cleft extensions of weak Hopf algebras.https://www.zbmath.org/1456.160322021-04-16T16:22:00+00:00"Guccione, Jorge A."https://www.zbmath.org/authors/?q=ai:guccione.jorge-alberto"Guccione, Juan J."https://www.zbmath.org/authors/?q=ai:guccione.juan-jose"Valqui, Christian"https://www.zbmath.org/authors/?q=ai:valqui.christianThe main motivation of this paper is to investigate the notion of clef extension for a weak Hopf algebra in a symmetric monoidal category with split idempotents. In Section 2, Definition 2.30, the authors introduce a notion of cleft extension for a weak Hopf algebra in the following way: Let \(H\) be a weak Hopf algebra, let \(B\) be a right \(H\)-comodule algebra and let \(j : A\rightarrow B\) be an algebra monomorphism. The pair \((B, j)\) is an extension of \(A\) by \(H\) if \(j\) is the equalizer of \(\rho_{B}\) and
\((B \otimes \Pi^{L})\circ \rho_{B}\) and we say that an extension \((B, j)\) is \(H\)-cleft if there exists a convolution invertible total integral \(\gamma:H\rightarrow B\). In this setting \(\rho_{B}\) denotes the coaction, \(\Pi^{L}\) the target morphism of \(H\) and for \(\gamma\) ``convolution invertible total integral'' means that \(\gamma\) is a morphism of right \(H\)-comodules, \(\gamma\circ \eta=\eta\) and there exists a morphism \(\gamma^{-1}\) satisfying the following identities:
\[\gamma^{-1}\ast \gamma=\gamma\circ \Pi^{R}, \;\; \gamma\ast \gamma^{-1}=\gamma\circ \Pi^{L}, \;\; \gamma^{-1}\ast \gamma\ast \gamma^{-1}=\gamma^{-1}.\]
This definition improves the one introduced in [\textit{J. N. Alonso Álvarez} et al., Chin. Ann. Math., Ser. B 35, No. 2, 161--190 (2014; Zbl 1301.18010)] and in the cocommutative setting are the same.
The paper is organized as follows: in Section 1 and 2 the authors recall and review the basic properties
of weak crossed products, introduced in [\textit{J. N. Alonso Álvarez} et al., Appl. Categ. Struct. 18, No. 3, 231--258 (2010; Zbl 1205.16025)] (see also [\textit{J. M. Fernández Vilaboa} et al., J. Pure Appl. Algebra 213, No. 12, 2244--2261 (2009; Zbl 1200.18002)]), and introduce the notion of cleft extension for a weak Hopf algebra. In Section 3, they extend the concept of equivalence of crossed products, introduced in [Alonso Álvarez et al. Zbl 1301.18010] for cocommutative Hopf algebras, to the setting of arbitrary weak bialgebras with a cocycle not invertible in general. In Section 4, the authors continue the study started in Section 2 about weak H-module algebras and in the fifth section they prove that each weak crossed product of a weak H-module algebra \(A\) by a weak Hopf algebra \(H\) with invertible cocycle, is an H-cleft extension of \(A\). In Section 6, for a weak Hopf algebra \(H\), they obtain that
each \(H\)-cleft extension is isomorphic to a weak crossed product with invertible cocycle
by a weak \(H\)-module algebra. Finally, under these conditions, the authors prove that the category of unitary weak crossed products of \(A\) by \(H\) with invertible cocycle, such that \(A\) is a weak \(H\)-module algebra, is equivalent to the category of \(H\)-cleft extensions of \(A\).
Reviewer: Ramón González Rodríguez (Vigo)Center of skew \textit{PBW} extensions.https://www.zbmath.org/1456.160382021-04-16T16:22:00+00:00"Lezama, Oswaldo"https://www.zbmath.org/authors/?q=ai:lezama.oswaldo"Venegas, Helbert"https://www.zbmath.org/authors/?q=ai:venegas.helbertTan's epsilon-determinant and ranks of matrices over semirings.https://www.zbmath.org/1456.150082021-04-16T16:22:00+00:00"Mohindru, Preeti"https://www.zbmath.org/authors/?q=ai:mohindru.preeti"Pereira, Rajesh"https://www.zbmath.org/authors/?q=ai:pereira.rajeshSummary: We use the \(\varepsilon\)-determinant introduced by \textit{Ya-Jia Tan} [Linear Multilinear Algebra 62, No. 4, 498--517 (2014; Zbl 1298.15014)] to define a family of ranks of matrices over certain semirings. We show that these ranks generalize some known rank functions over semirings such as the determinantal rank. We also show that this family of ranks satisfies the rank-sum and Sylvester inequalities. We classify all bijective linear maps which preserve these ranks.\(A_\infty \)-structures associated with pairs of 1-spherical objects and noncommutative orders over curves.https://www.zbmath.org/1456.140232021-04-16T16:22:00+00:00"Polishchuk, Alexander"https://www.zbmath.org/authors/?q=ai:polishchuk.alexander-eSummary: We show that pairs \((X,Y)\) of 1-spherical objects in \(A_\infty \)-categories, such that the morphism space \(\operatorname{Hom}(X,Y)\) is concentrated in degree 0, can be described by certain noncommutative orders over (possibly stacky) curves. In fact, we establish a more precise correspondence at the level of isomorphism of moduli spaces which we show to be affine schemes of finite type over \(\mathbb{Z} \).On Magnus' Freiheitssatz and free polynomial algebras.https://www.zbmath.org/1456.200272021-04-16T16:22:00+00:00"Fine, Benjamin"https://www.zbmath.org/authors/?q=ai:fine.benjamin-l"Kreuzer, Martin"https://www.zbmath.org/authors/?q=ai:kreuzer.martin"Rosenberger, Gerhard"https://www.zbmath.org/authors/?q=ai:rosenberger.gerhardSummary: The Freiheitssatz of Magnus for one-relator groups is one of the cornerstones of combinatorial group theory. In this short note which is mostly expository we discuss the relationship between the Freiheitssatz and corresponding results in free power series rings over fields. These are related to results of Schneerson not readily available in English. This relationship uses a faithful representation of free groups due to Magnus. Using this method in free polynomial algebras provides a proof of the Freiheitssatz for one-relation monoids. We show how the classical Freiheitssatz depends on a condition on certain ideals in power series rings in noncommuting variables over fields. A proof of this result over fields would provide a completely dif erent proof of the classical Freiheitssatz.On free Gelfand-Dorfman-Novikov-Poisson algebras and a PBW theorem.https://www.zbmath.org/1456.170012021-04-16T16:22:00+00:00"Bokut, L. A."https://www.zbmath.org/authors/?q=ai:bokut.leonid-a"Chen, Yuqun"https://www.zbmath.org/authors/?q=ai:chen.yuqun"Zhang, Zerui"https://www.zbmath.org/authors/?q=ai:zhang.zeruiGelfand-Dorfman-Novikov algebras (GDN algebras, known also as Novikov algebras) were introduced independently in [\textit{I. M. Gel'fand} and \textit{I. Ya. Dorfman}, Funkts. Anal. Prilozh. 13, No. 4, 13--30 (1979; Zbl 0428.58009); translation in Funct. Anal. Appl. 13, 248--262 (1980; Zbl 0437.58009)]
in connection with Hamiltonian operators in the formal calculus of variations and in [\textit{A. A. Balinskii} and \textit{S. P. Novikov}, Sov. Math., Dokl. 32, 228--231 (1985; Zbl 0606.58018); translation from Dokl. Akad. Nauk SSSR 283, 1036--1039 (1985)] in connection with linear Poisson brackets of hydrodynamic type.
These algebras satisfy the identities of left symmetry
\[ x\circ(y\circ z)-(x\circ y)\circ z = y\circ (x\circ z)-(y\circ x)\circ z\]
and right commutativity
\[ (x\circ y)\circ z = (x\circ z)\circ y. \]
GDN-Poisson algebras were introduced in [\textit{X. Xu}, J. Algebra 185, No. 3, 905--934 (1996; Zbl 0863.17003); J. Algebra 190, No. 2, 253--279 (1997; Zbl 0872.17030)]. These are GDN algebras with an additional operation \(\cdot\) which equips the algebra with the structure of a commutative associative algebra with the compatibility conditions
\[ (x\cdot y)\circ z = x\cdot (y\circ z)\quad\text{and}\quad
(x\circ y)\cdot z-x\circ (y\cdot z) = (y\circ x)\cdot z-y\circ (x\cdot z). \]
In the paper under review the authors construct a linear basis of the free GDN-Poisson algebra.
Then they define the notion of a special GDN-Poisson admissible algebra. This is a differential algebra with two commutative associative products and some extra identities.The authors prove that any GDN-Poisson algebra can be embedded into its universal enveloping special GDN-Poisson admissible algebra.
Finally, they establish that any GDN-Poisson algebra with the identity
\[ x\circ(y\cdot z)=(x\circ y)\cdot z+(x\circ z)\cdot y\]
is isomorphic to a commutative associative differential algebra both as GDN-Poisson algebra and as a commutative associative differential algebra.
Reviewer: Vesselin Drensky (Sofia)Representing nilpotent matrices as single commutators.https://www.zbmath.org/1456.160252021-04-16T16:22:00+00:00"Hoopes-Boyd, Emily"https://www.zbmath.org/authors/?q=ai:hoopes-boyd.emilySummary: We will show that every nilpotent element \(N\) in \(M_n(D)\), the ring of square matrices over a division ring, can be presented as a single commutator, that is, \(N=AB-BA\) for some matrices \(A, B\) in \(M_n(D)\). We will also construct an example illustrating that there exists a prime ring with unity over which some nilpotent matrices cannot be presented as commutators.Total graph of a commutative semiring with respect to singular ideal.https://www.zbmath.org/1456.130102021-04-16T16:22:00+00:00"Goswami, Nabanita"https://www.zbmath.org/authors/?q=ai:goswami.nabanita"Saikia, Helen K."https://www.zbmath.org/authors/?q=ai:saikia.helen-kumariTotal graphs of commutative rings with respect to various subsets are studied by several authors in recent times. In parallel to this, for a commutative semiring \(S,\) in this paper, the authors define the total graph of \(S\) with respect to its singular ideal \(Z(S)\). This graph is defined as the undirected graph \(T(\Gamma(S))\) with \(S\) as the vertex set and two vertices \(x\) and \(y(x\neq y)\) are adjacent if \(x+y \in Z(S)\). Having defined this graph, the authors obtain some characterization of the same and also study the interplay between algebraic properties of \(S\) and graph theoretic properties of \(T(\Gamma(S)).\) For any subset \(A\) of \(S\), the induced subgraph of \(T(\Gamma(S))\) with all elements of \(A\) as vertices is denoted by \(T(\Gamma(A)).\) They discuss some properties of \(T(Γ(\overline{Z(S)}))\) where \(\overline{Z(S)} = S \setminus Z(S)\).
Reviewer: T. Tamizh Chelvam (Tirunelveli)Algebras of generalized dihedral type.https://www.zbmath.org/1456.160082021-04-16T16:22:00+00:00"Erdmann, Karin"https://www.zbmath.org/authors/?q=ai:erdmann.karin"Skowroński, Andrzej"https://www.zbmath.org/authors/?q=ai:skowronski.andrzejSummary: We provide a complete classification of all algebras of generalized dihedral type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.Representation type of surfaces in \(\mathbb{P}^3\).https://www.zbmath.org/1456.140212021-04-16T16:22:00+00:00"Ballico, Edoardo"https://www.zbmath.org/authors/?q=ai:ballico.edoardo"Huh, Sukmoon"https://www.zbmath.org/authors/?q=ai:huh.sukmoonA possible way to measure the complexity of a given \(n\)-dimensional polarized variety \((X, \mathcal{O}_X (1))\) is to ask for the families of non-isomorphic indecomposable aCM (arithmetically Cohen-Macaulay) vector bundles that it supports (recall that a vector bundle \(\mathcal{E}\) on \(X\) is aCM if \(H^i(X,\mathcal{E}\otimes\mathcal{O}_X(t))= 0\) for all \(t\in\mathbb{Z}\) and \(i=1,\dots, n-1\)). The first result on this direction was Horrocks' theorem which states that on the projective space the only indecomposable aCM bundle up to twist is the structure sheaf \(\mathcal{O}_{\mathbb{P}^n}\).
Inspired by analogous classifications in quiver theory and representation theory, a classification of polarized varieties as \textit{finite, tame and wild} was proposed. ACM varieties of finite type (namely, supporting only a finite number of non-isomorphic indecomposable aCM vector bundles) were completely classified in [\textit{ D. Eisenbud} and \textit{J. Herzog}, Math. Ann. 280, No. 2, 347--352 (1988; Zbl 0616.13011]. If we look at the other extreme of complexity we would find the varieties of wild representation type, namely, varieties for which there exist \(r\)-dimensional families of non-isomorphic indecomposable aCM bundles for arbitrary large \(r\). Recently, the representation type of any reduce aCM polarized variety has been determined [\textit{D. Faenzi} and \textit{J. Pons-Llopis}, ``The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties'', Preprint, \url{arXiv:1504.03819}].
In the article under review, the authors prove that every surface \(X\) with a regular point in the three-dimensional projective space of degree at least four is of wild representation type under the condition that either \(X\) is integral or Pic\((X)\) is \(\mathbb{Z}\)-generated by \(\mathcal{O}_X(1)\). Alongside, they also prove the interesting result that every non-integral aCM variety of dimension at least two is also very wild: namely there exist arbitrarily large dimensional families of pairwise non-isomorphic aCM non-locally free sheaves of rank one.
Reviewer: Joan Pons-Llopis (Maó)Results related to self-injectivity of the group ring.https://www.zbmath.org/1456.160222021-04-16T16:22:00+00:00"Schwiebert, Ryan C."https://www.zbmath.org/authors/?q=ai:schwiebert.ryan-cSummary: From the early 1960s to the early 1970s, there was much activity leading up to a proof that if the group ring \(R[G]\) is right self-injective, then \(G\) is finite. Unfortunately, it is difficult to find this fact both stated and proven in print and in full generality. This article presents new renditions of the main proofs, and chronicles what the author learned while sorting out the literature for this period.