Recent zbMATH articles in MSC 16G https://www.zbmath.org/atom/cc/16G 2021-04-16T16:22:00+00:00 Werkzeug Higher derivations of finitary incidence algebras. https://www.zbmath.org/1456.16026 2021-04-16T16:22:00+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.feng A 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) Matrix method for persistence modules on commutative ladders of finite type. https://www.zbmath.org/1456.55004 2021-04-16T16:22:00+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.hiroshi A 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) Auslander-Reiten quiver and representation theories related to KLR-type Schur-Weyl duality. https://www.zbmath.org/1456.16010 2021-04-16T16:22:00+00:00 "Oh, Se-jin" https://www.zbmath.org/authors/?q=ai:oh.se-jin Summary: 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 $$U_q(E_{6,7,8}^{(1)})$$. Combinatorics of exceptional sequences in type A. https://www.zbmath.org/1456.16009 2021-04-16T16:22:00+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.jonah Summary: 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. Algebras of generalized dihedral type. https://www.zbmath.org/1456.16008 2021-04-16T16:22:00+00:00 "Erdmann, Karin" https://www.zbmath.org/authors/?q=ai:erdmann.karin "Skowroński, Andrzej" https://www.zbmath.org/authors/?q=ai:skowronski.andrzej Summary: 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. Symmetries of the simply-laced quantum connections and quantisation of quiver varieties. https://www.zbmath.org/1456.81272 2021-04-16T16:22:00+00:00 "Rembado, Gabriele" https://www.zbmath.org/authors/?q=ai:rembado.gabriele Summary: 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. Representation type of surfaces in $$\mathbb{P}^3$$. https://www.zbmath.org/1456.14021 2021-04-16T16:22:00+00:00 "Ballico, Edoardo" https://www.zbmath.org/authors/?q=ai:ballico.edoardo "Huh, Sukmoon" https://www.zbmath.org/authors/?q=ai:huh.sukmoon A 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) Relative cluster tilting objects in triangulated categories. https://www.zbmath.org/1456.16011 2021-04-16T16:22:00+00:00 "Yang, Wuzhong" https://www.zbmath.org/authors/?q=ai:yang.wuzhong "Zhu, Bin" https://www.zbmath.org/authors/?q=ai:zhu.bin Let $$\mathcal{D}$$ 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 $$T$$-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 $$T$$-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 $$T$$-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) On the Morita reduced versions of skew group algebras of path algebras. https://www.zbmath.org/1456.16023 2021-04-16T16:22:00+00:00 "Le Meur, Patrick" https://www.zbmath.org/authors/?q=ai:le-meur.patrick Summary: 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. Analogues of centralizer subalgebras for fiat 2-categories and their 2-representations. https://www.zbmath.org/1456.18016 2021-04-16T16:22:00+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.xiaoting Finitary 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) Cyclic cohomology and Chern-Connes pairing of some crossed product algebras. https://www.zbmath.org/1456.58008 2021-04-16T16:22:00+00:00 "Quddus, Safdar" https://www.zbmath.org/authors/?q=ai:quddus.safdar Summary: 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. Higher Auslander algebras of type $$\mathbb{A}$$ and the higher Waldhausen $$\mathsf{S}$$-constructions. https://www.zbmath.org/1456.18011 2021-04-16T16:22:00+00:00 "Jasso, Gustavo" https://www.zbmath.org/authors/?q=ai:jasso.gustavo Summary: 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 combinatorics of gentle algebras. https://www.zbmath.org/1456.16007 2021-04-16T16:22:00+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.emine Summary: 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.