The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach

Piotr Dowbor; Andrzej Mróz

Displaying similar documents to “The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach”

The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras

Piotr Dowbor, Andrzej Mróz (2008)

Colloquium Mathematicae

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Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m (M) = {(m_{x})}_{x \in X} \in ℕ^{X}$ such that $M ≅ ⨁_{x \in X} X_{x}^{m_{x}}$ is studied. A precise formula for $d i m_{k} H o m_{Λ} (M, X)$ , for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors...

Standardly stratified split and lower triangular algebras

Eduardo do N. Marcos, Hector A. Merklen, Corina Sáenz (2002)

Colloquium Mathematicae

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In the first part, we study algebras A such that A = R ⨿ I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for $A = [\begin{matrix} U & 0 \\ M & V \end{matrix}]$ , we show that A is standardly stratified if and only if the algebra R = U × V is standardly stratified and $_{V} M$ is a good V-module.

A representation theorem for tense $n \times m$ -valued Łukasiewicz-Moisil algebras

Aldo Victorio Figallo, Gustavo Pelaitay (2015)

Mathematica Bohemica

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In 2000, Figallo and Sanza introduced $n \times m$ -valued Łukasiewicz-Moisil algebras which are both particular cases of matrix Łukasiewicz algebras and a generalization of $n$ -valued Łukasiewicz-Moisil algebras. Here we initiate an investigation into the class $_{n \times m}$ of tense $n \times m$ -valued Łukasiewicz-Moisil algebras (or tense LM $_{n \times m}$ -algebras), namely $n \times m$ -valued Łukasiewicz-Moisil algebras endowed with two unary operations called tense operators. These algebras constitute a generalization of tense...

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...

On some properties of the upper central series in Leibniz algebras

Leonid A. Kurdachenko, Javier Otal, Igor Ya. Subbotin (2019)

Commentationes Mathematicae Universitatis Carolinae

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This article discusses the Leibniz algebras whose upper hypercenter has finite codimension. It is proved that such an algebra $L$ includes a finite dimensional ideal $K$ such that the factor-algebra $L / K$ is hypercentral. This result is an extension to the Leibniz algebra of the corresponding result obtained earlier for Lie algebras. It is also analogous to the corresponding results obtained for groups and modules.

On generalized Beurling’s theorem and symmetry of $L_{1}$ -group algebras

Zenobia Anusiak (1971)

Colloquium Mathematicae

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Category $𝒪$ for quantum groups

Henning Haahr Andersen, Volodymyr Mazorchuk (2015)

Journal of the European Mathematical Society

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In this paper we study the BGG-categories $𝒪_{q}$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $𝒪$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $𝒪$ and for finite dimensional $U_{q}$ -modules we are able...

Derived endo-discrete artin algebras

Raymundo Bautista (2006)

Colloquium Mathematicae

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Let Λ be an artin algebra. We prove that for each sequence ${(h_{i})}_{i \in ℤ}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X \in^{b} (Λ)$ , the bounded derived category of Λ, with $l e n g t h_{E (X)} H^{i} (X) = h_{i}$ for all i ∈ ℤ and E(X) the endomorphism ring of X in $^{b} (Λ)$ if and only if $^{b} (M o d Λ)$ , the bounded derived category of the category $M o d Λ$ of all left Λ-modules, has no generic objects in the sense of [4].

On the structure theory of the Iwasawa algebra of a p-adic Lie group

Otmar Venjakob (2002)

Journal of the European Mathematical Society

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This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, $Λ$ of a $p$ -adic analytic group $G$ . For $G$ without any $p$ -torsion element we prove that $Λ$ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null $Λ$ -module. This is classical when $G = ℤ_{p}^{k}$ for some integer $k \geq 1$ , but was previously unknown in the non-commutative case. Then the category...

On cardinalities of algebras of formulas for $ω_{0}$ -categorical theories

Jan Waszkiewicz (1973)

Colloquium Mathematicae

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Top-stable and layer-stable degenerations and hom-order

S. O. Smalø, A. Valenta (2007)

Colloquium Mathematicae

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Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration $M <_{d e g} N$ such that $^{t} M /^{t + 1} M ≃^{t} N /^{t + 1} N$ for all t. Given a module M with square-free top and a projective cover P, she showed that $d i m_{k} H o m (M, M) = d i m_{k} H o m (P, M)$ if and only if M has no proper degeneration $M <_{d e g} N$ where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from...

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.

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 $8$ .

Stratified modules over an extension algebra

Erzsébet Lukács, András Magyar (2018)

Czechoslovak Mathematical Journal

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Let $A$ be a standard Koszul standardly stratified algebra and $X$ an $A$ -module. The paper investigates conditions which imply that the module ${Ext}_{A}^{*} (X)$ over the Yoneda extension algebra $A^{*}$ is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of $A$ is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

Coalgebras, comodules, pseudocompact algebras and tame comodule type

Daniel Simson (2001)

Colloquium Mathematicae

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We develop a technique for the study of K-coalgebras and their representation types by applying a quiver technique and topologically pseudocompact modules over pseudocompact K-algebras in the sense of Gabriel [17], [19]. A definition of tame comodule type and wild comodule type for K-coalgebras over an algebraically closed field K is introduced. Tame and wild coalgebras are studied by means of their finite-dimensional subcoalgebras. A weak version of the tame-wild dichotomy theorem of...

(Strongly) Gorenstein injective modules over upper triangular matrix Artin algebras

Chao Wang, Xiao Yan Yang (2017)

Czechoslovak Mathematical Journal

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Let $Λ = (\begin{matrix} A & M \\ 0 & B \end{matrix})$ be an Artin algebra. In view of the characterization of finitely generated Gorenstein injective $Λ$ -modules under the condition that $M$ is a cocompatible $(A, B)$ -bimodule, we establish a recollement of the stable category $\bar{Ginj (Λ)}$ . We also determine all strongly complete injective resolutions and all strongly Gorenstein injective modules over $Λ$ .

Some Results About Holonomic $ε$ -Modules

Jan-Erik Björk (1983)

Recherche Coopérative sur Programme n°25

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Relative tilting modules with respect to a semidualizing module

Maryam Salimi (2019)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, and let $C$ be a semidualizing $R$ -module. The notion of $C$ -tilting $R$ -modules is introduced as the relative setting of the notion of tilting $R$ -modules with respect to $C$ . Some properties of tilting and $C$ -tilting modules and the relations between them are mentioned. It is shown that every finitely generated $C$ -tilting $R$ -module is $C$ -projective. Finally, we investigate some kernel subcategories related to $C$ -tilting modules.