The component quiver of a self-injective artin algebra

Alicja Jaworska; Andrzej Skowroński

Displaying similar documents to “The component quiver of a self-injective artin algebra”

The duality of Auslander-Reiten quiver of path algebras

Bo Hou, Shilin Yang (2019)

Czechoslovak Mathematical Journal

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Let $Q$ be a finite union of Dynkin quivers, $G \subseteq Aut (𝕜 Q)$ a finite abelian group, $\hat{Q}$ the generalized McKay quiver of $(Q, G)$ and $Γ_{Q}$ the Auslander-Reiten quiver of $𝕜 Q$ . Then $G$ acts functorially on the quiver $Γ_{Q}$ . We show that the Auslander-Reiten quiver of $𝕜 \hat{Q}$ coincides with the generalized McKay quiver of $(Γ_{Q}, G)$ .

Trivialization of $𝒞 (X)$ -algebras with strongly self-absorbing fibres

Marius Dadarlat, Wilhelm Winter (2008)

Bulletin de la Société Mathématique de France

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Suppose $A$ is a separable unital $𝒞 (X)$ -algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$ -injective $C^{*}$ -algebra $𝒟$ . We show that $A$ and $𝒞 (X) \otimes 𝒟$ are isomorphic as $𝒞 (X)$ -algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

Transfer of derived equivalences from subalgebras to endomorphism algebras II

Shengyong Pan, Jiahui Yu (2024)

Czechoslovak Mathematical Journal

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We investigate derived equivalences between subalgebras of some $Φ$ -Auslander-Yoneda algebras from a class of $n$ -angles in weakly $n$ -angulated categories. The derived equivalences are obtained by transferring subalgebras induced by $n$ -angles to endomorphism algebras induced by approximation sequences. Then we extend our constructions in T. Brüstle, S. Y. Pan (2016) to $n$ -angle cases. Finally, we give an explicit example to illustrate our result.

Does the endomorphism poset $P^{P}$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978

Jonathan David Farley (2023)

Mathematica Bohemica

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Duffus wrote in his 1978 Ph.D. thesis, “It is not obvious that $P$ is connected and $P^{P} ≅ Q^{Q}$ imply that $Q$ is connected”, where $P$ and $Q$ are finite nonempty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P ≅ Q$ .

Stable tubes in extriangulated categories

Li Wang, Jiaqun Wei, Haicheng Zhang (2022)

Czechoslovak Mathematical Journal

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Let $𝒳$ be a semibrick in an extriangulated category. If $𝒳$ is a $τ$ -semibrick, then the Auslander-Reiten quiver $Γ (ℱ (𝒳))$ of the filtration subcategory $ℱ (𝒳)$ generated by $𝒳$ is $ℤ 𝔸_{\infty}$ . If $𝒳 = {X_{i}}_{i = 1}^{t}$ is a $τ$ -cycle semibrick, then $Γ (ℱ (𝒳))$ is $ℤ 𝔸_{\infty} / τ_{𝔸}^{t}$ .

On D-connected sets in the space $V^{q}$

S. Topa (1976)

Annales Polonici Mathematici

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A representation theorem for $(X_{1} - 1) (X_{2} - 1) . . . (X_{n} - 1)$ and its applications

D. Ž. Djoković (1969)

Annales Polonici Mathematici

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$L_{p, q}$ spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p, q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p, q} (B)$ ........ 235. Examples in $L_{p, q}$ theory...................................................................................

Integral Representation of States on a $C^{*}$ -Algebra

David Ruelle (1969)

Recherche Coopérative sur Programme n°25

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Integral Representation of States on a $C^{*}$ -Algebra

David Ruelle (1973)

Recherche Coopérative sur Programme n°25

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On twisted group algebras of OTP representation type

Leonid F. Barannyk, Dariusz Klein (2012)

Colloquium Mathematicae

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Assume that S is a commutative complete discrete valuation domain of characteristic p, S* is the unit group of S and $G = G_{p} \times B$ is a finite group, where $G_{p}$ is a p-group and B is a p’-group. Denote by $S^{λ} G$ the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). We give necessary and sufficient conditions for $S^{λ} G$ to be of OTP representation type, in the sense that every indecomposable $S^{λ} G$ -module is isomorphic to the outer tensor product V W of an indecomposable $S^{λ} G_{p}$ -module V and an irreducible...

Bicrossed products of generalized Taft algebra and group algebras

Dingguo Wang, Xiangdong Cheng, Daowei Lu (2022)

Czechoslovak Mathematical Journal

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Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m, d} (q) ⋈ k [G]$ between the generalized Taft algebra $H_{m, d} (q)$ and group algebra $k [G]$ is actually the smash product $H_{m, d} (q) ♯ k [G]$ . Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$ . As an application, the classification of $H_{m, d} (q) ⋈ k [C_{n_{1}} \times C_{n_{2}}]$ is completely presented by generators and relations, where $C_{n}$ denotes the $n$ -cyclic group.

On twisted group algebras of OTP representation type over the ring of p-adic integers

Leonid F. Barannyk, Dariusz Klein (2016)

Colloquium Mathematicae

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Let $ℤ ̂_{p}$ be the ring of p-adic integers, $U (ℤ ̂_{p})$ the unit group of $ℤ ̂_{p}$ and $G = G_{p} \times B$ a finite group, where $G_{p}$ is a p-group and B is a p’-group. Denote by $ℤ ̂_{p}^{λ} G$ the twisted group algebra of G over $ℤ ̂_{p}$ with a 2-cocycle $λ \in Z ² (G, U (ℤ ̂_{p}))$ . We give necessary and sufficient conditions for $ℤ ̂_{p}^{λ} G$ to be of OTP representation type, in the sense that every indecomposable $ℤ ̂_{p}^{λ} G$ -module is isomorphic to the outer tensor product V W of an indecomposable $ℤ ̂_{p}^{λ} G_{p}$ -module V and an irreducible $ℤ ̂_{p}^{λ} B$ -module W.

Homological dimensions for endomorphism algebras of Gorenstein projective modules

Aiping Zhang, Xueping Lei (2024)

Czechoslovak Mathematical Journal

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Let $A$ be a CM-finite Artin algebra with a Gorenstein-Auslander generator $E$ , $M$ be a Gorenstein projective $A$ -module and $B = {End}_{A} M$ . We give an upper bound for the finitistic dimension of $B$ in terms of homological data of $M$ . Furthermore, if $A$ is $n$ -Gorenstein for $2 \leq n < \infty$ , then we show the global dimension of $B$ is less than or equal to $n$ plus the $B$ -projective dimension of ${Hom}_{A} (M, E) .$ As an application, the global dimension of ${End}_{A} E$ is less than or equal to $n$ .

Matrix representation of finite effect algebras

Grzegorz Bińczak, Joanna Kaleta, Andrzej Zembrzuski (2023)

Kybernetika

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In this paper we present representation of finite effect algebras by matrices. For each non-trivial finite effect algebra $E$ we construct set of matrices $M (E)$ in such a way that effect algebras $E_{1}$ and $E_{2}$ are isomorphic if and only if $M (E_{1}) = M (E_{2})$ . The paper also contains the full list of matrices representing all nontrivial finite effect algebras of cardinality at most $8$ .

The moduli space of commutative algebras of finite rank

Bjorn Poonen (2008)

Journal of the European Mathematical Society

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The moduli space of rank- $n$ commutative algebras equipped with an ordered basis is an affine scheme $𝔅_{n}$ of finite type over $ℤ$ , with geometrically connected fibers. It is smooth if and only if $n \leq 3$ . It is reducible if $n \geq 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$ ). The relative dimension of $𝔅_{n}$ is $\frac{2}{27} n^{3} + O (n^{8 / 3})$ . The subscheme parameterizing étale algebras is isomorphic to ${GL}_{n} / S_{n}$ , which is of dimension only $n^{2}$ . For $n \geq 8$ , there exist algebras that are not limits of étale algebras. The dimension...

Some results on $(n, d)$ -injective modules, $(n, d)$ -flat modules and $n$ -coherent rings

Zhanmin Zhu (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $n, d$ be two non-negative integers. A left $R$ -module $M$ is called $(n, d)$ -injective, if ${Ext}^{d + 1} (N, M) = 0$ for every $n$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called $(n, d)$ -flat, if ${Tor}_{d + 1} (V, N) = 0$ for every $n$ -presented left $R$ -module $N$ . A left $R$ -module $M$ is called weakly $n$ - $F P$ -injective, if ${Ext}^{n} (N, M) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called weakly $n$ -flat, if ${Tor}_{n} (V, N) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . In this paper, we give some characterizations and properties of $(n, d)$ -injective modules and $(n, d)$ -flat modules in...

Higher-dimensional Auslander-Reiten sequences

Jiangsha Li, Jing He (2024)

Czechoslovak Mathematical Journal

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Zhou and Zhu have shown that if $𝒞$ is an $(n + 2)$ -angulated category and $𝒳$ is a cluster tilting subcategory of $𝒞$ , then the quotient category $𝒞 / 𝒳$ is an $n$ -abelian category. We show that if $𝒞$ has Auslander-Reiten $(n + 2)$ -angles, then $𝒞 / 𝒳$ has Auslander-Reiten $n$ -exact sequences.

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{s g}, I^{s g})$ and $(Q^{g}, I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K Q^{s g} / ⟨ I^{s g} ⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K Q^{g} / ⟨ I^{g} ⟩$ . As a corollary, we find that $g l d i m K Q^{s g} / ⟨ I^{s g} ⟩ < \infty$ if and only if $g l d i m K Q / ⟨ I ⟩ < \infty$ if and only if $g l d i m K Q^{g} / ⟨ I^{g} ⟩ < \infty$ .