Some Hall polynomials for representation-finite trivial extension algebras
Let \(k\) be a finite field, \(\Lambda\) a finite dimensional associative \(k\)-algebra and denote by \(\text{mod }\Lambda\) the category of finitely generated right \(\Lambda\)-modules. For \(M,L,N\in\text{mod }\Lambda\) let \(F^M_{LN}\) be the number of submodules \(U\) of \(M\) such that \(U\simeq N\) and \(M/U\simeq L\). Let \(\Omega\) be the set of isomorphism classes of finite field extensions \(E\) of \(k\) such that for any indecomposable \(\Lambda\)-module \(M\), \(\text{End }M/\text{rad End }M\) is a field. Assuming that \(\Omega\) is infinite, one says, according to \textit{C. M. Ringel} [in Lond. Math. Soc. Lect. Note Ser. 168, 284-291 (1992; Zbl 0764.17005)], that \(\Lambda\) has Hall polynomials if for any \(X,Y,Z\in\text{mod }\Lambda\), there is \(\phi^Y_{ZX}\in\mathbb{Z}[T]\) such that \(\phi_{ZX}^Y(|E|)=F^{Y^E}_{Z^EX^E}\) for any \(E\in\Omega,\) where \(X^E=X\otimes_k E\). The main result of the paper states that if \(\Lambda\) is a representation finite trivial extension algebra, then for any \(L,M,N\in\text{mod }\Lambda\) with \(N\) indecomposable, \(\Lambda\) has Hall polynomials \(\phi^M_{LN}\) and \(\phi^M_{NL}\). Using these polynomials it is then shown that the free abelian group with basis \(\text{ind }\Lambda\) can be structured into a Lie algebra which is related to the degenerated Hall algebra of \(\Lambda\).
categories of finitely generated modules
scientific article; zbMATH DE number 1097698
scientific article