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finite-dimensional algebras

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Let ${\mathcal {A}}$ be a unital ${\mathbb {F}}$ -algebra and let ${\mathcal {S}}$ be a generating set of ${\mathcal {A}}$ . The length of ${\mathcal {S}}$ is the smallest number k such that ${\mathcal {A}}$ equals the ${\mathbb {F}}$ -linear span of all products of length at most k of elements from ${\mathcal {S}}$ . The length of ${\mathcal {A}}$ , denoted by $l({\mathcal {A}})$ , is defined to be the maximal length of its generating sets. We show that $l({\mathcal {A}})$ does not exceed the maximum of $\dim {\mathcal {A}} / 2$ and $m({\mathcal {A}})-1$ , where $m({\mathcal {A}})$ is the largest degree of the minimal polynomial among all elements of the algebra ${\mathcal {A}}$ . As an application, we show that for arbitrary odd n, the length of the group algebra of the dihedral group of order $2n$ equals n.

1 - Auslander–Reiten Theory of Finite-Dimensional Algebras
Edited by David Jordan, University of Edinburgh, Nadia Mazza, Lancaster University, Sibylle Schroll, Universität zu Köln
Book:

Modern Trends in Algebra and Representation Theory

Published online:

25 November 2023

Print publication:

17 August 2023, pp 5-45
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Summary

We give a brief introduction to the representation theory of finite-dimensional algebras via quivers and Auslander–Reiten theory. We describe the knitting algorithm, which gives a way to compute the Auslander–Reiten quiver of many algebras of finite-representation type. We also present recent work on the description of the representation theory of gentle and skew-gentle algebras via the geometry of oriented surfaces with boundary. An application of these geometric models is given in the context of τ-tilting theory.

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AN UPPER BOUND ON THE LENGTH OF AN ALGEBRA AND ITS APPLICATION TO THE GROUP ALGEBRA OF THE DIHEDRAL GROUP

1 - Auslander–Reiten Theory of Finite-Dimensional Algebras

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AN UPPER BOUND ON THE LENGTH OF AN ALGEBRA AND ITS APPLICATION TO THE GROUP ALGEBRA OF THE DIHEDRAL GROUP

1 - Auslander–Reiten Theory of Finite-Dimensional Algebras

Summary