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2 votes
0 answers
70 views

Rigid modules for hereditary algebras

Let $A=KQ$ be a path algebra of a connected quiver. (K algebraically closed if it helps) Question: Is there an explicit classification of all indecomposable $A$-modules $M$ that are rigid, that is $...
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2 votes
0 answers
91 views

When does a stable endomorphism ring have injective dimension at most one?

tLet $A$ be a Frobenius algebra (we can assume that $A$ is given by quiver and relations) and let $M$ be a basic $A$-module without projective direct summands (we can assume we know the decomposition ...
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5 votes
0 answers
142 views

A practical way to check whether a module is periodic

A module $M$ over a finite dimensional selfinjective algebra $A$ over a field $K$ is called periodic if $M \cong \Omega^n(M)$ for some $n \geq 1$. We assume here that $M$ is simple and that A is a ...
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3 votes
0 answers
78 views

Quiver algebras of Dynkin type

Let $kQ$ be one of the Dynin path algebras of type $A_n , D_n $ or $E_i$ for $i=6,7,8$. Question 1: How many (up to isomorphism) quiver algebras are there that are derived equivalent to $kQ$? ...
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4 votes
0 answers
241 views

Finding local algebra and relations lottery

This can be seen as an attempt for a mini Polymath project on homological properties of (local) finite dimensional algebras. You only need to know what a finite dimensional algebra is and have GAP to ...
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4 votes
0 answers
155 views

Commutative algebras associated to simple Lie algebras

In Section 2 of the article https://www.sciencedirect.com/science/article/pii/S0021869307000385, the authors study the center $Z=Z_Q$ of certain preprojective like algebras associated to the simply ...
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11 votes
0 answers
202 views

Quiver and relations for blocks of category $\mathcal{O}$

In Vybornov - Perverse sheaves, Koszul IC-modules, and the quiver for the category $\mathscr O$ an algorithm is presented to calculate quiver and relations for blocks of category $\mathcal{O}$ . ...
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3 votes
1 answer
98 views

Finding automorphisms and cyclic modules via QPA

Given a symmetric finite dimensional algebra $A$ over a finite field with enveloping algebra $A$. Assume we know that $\Omega_{A^e}^i(A) \cong A_{f}$, where $f$ is some automorphism of the algebra $A$....
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8 votes
1 answer
193 views

Maximal numbers of summands in middle terms of short exact sequences

Let $A$ be a finite dimensional algebra and $M$ and $N$ indecomposable $A$-modules. Denote by $\xi(M,N)$ the maximal number of indecomposable summands of a modules $X$ such that there is a non-split ...
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2 votes
0 answers
45 views

On monomial and $\Omega^d$-finite algebras

Let $Q$ be a finite quiver and $I$ a monomial admissible ideal of the path algebra $KQ$ for a field $K$. Then an algebra $A=KQ/I$ is called a monomial algebra. It is well known that monomial algebras ...
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4 votes
1 answer
149 views

Testing whether a module generates $K_0(\mbox{mod-}A)$

Given a representation-finite (connected) quiver algebra $A$ and a module $M$. Is there a good way to test whether the set $\{ [N] \mid N \in \mathrm{add}(M) \}$ generates $K_0(\mbox{mod-}A)$? Can ...
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