On Frobenius algebras. (Q1415353)

The paper under review studies Frobenius factor algebras of finite dimensional algebras. Let \(D(\Lambda)\) be the \(K\)-dual of a finite dimensional \(K\)-algebra \(\Lambda\). Then \(D(\Lambda)\) becomes naturally a left and right \(\Lambda\)-module. Let \({\mathcal Q}_\Lambda\) be the set of all elements \(f\in D(\Lambda)\) whose left and right \(\Lambda\)-annihilator coincide. It is shown that a factor algebra \(\Lambda/I\) is Frobenius if and only if \(I=\text{Ann}_\Lambda(f)\) for some \(f\in{\mathcal Q}_\Lambda\). Hence the Frobenius factor algebras \(\Lambda/I\) are indexed by elements \(f\in {\mathcal Q}_\Lambda\) and denoted by \(\Lambda_f\). Moreover the group of units \(\Lambda^\times\) of \(\Lambda\) acts on \({\mathcal Q}_\Lambda\) and the orbit space \({\mathcal Q}_\Lambda/\Lambda^\times\) is shown to be in bijection to the set \({\mathcal F}_\Lambda\) of ideals \(I\) of \(\Lambda\) such that \(\Lambda/I\) is Frobenius. Given an automorphism \(\sigma\) of \(\Lambda\) the set \({\mathcal Q}_\Lambda(\sigma)=D(\Lambda/[\Lambda,\Lambda]_\sigma)\) is shown to be a subset of \({\mathcal Q}_\Lambda\). The union of all those subsets \({\mathcal Q}_\Lambda^*\) is in general a proper subset of \({\mathcal Q}_\Lambda\) as the author points out. Two Frobenius factor algebras \(\Lambda_f\) and \(\Lambda_g\) are said to be lifting-equivalent if there exists an isomorphism between \(\Lambda_f\) and \(\Lambda_g\) that can be lifted to an automorphism of \(\Lambda\). The author considers the question when isomorphic Frobenius factors are lifting-equivalent. For that the author considers the semi-direct product \({\mathcal G}_\Lambda:=\Aut_K(\Lambda)\propto\Lambda^\times\) and its action on \({\mathcal Q}_\Lambda\) resp. \({\mathcal Q}_\Lambda^*\). It is shown that the \({\mathcal G}_\Lambda\)-orbits of elements of \({\mathcal Q}_\Lambda\) are in bijection to the lifting-equivalent classes of Frobenius factors, i.e. there exists a bijection between the orbit spaces \({\mathcal Q}_\Lambda/{\mathcal G}_\Lambda\) and the set of lifting-equivalent classes \({\mathcal F}_\Lambda/\sim\). The main theorem of the paper states that for a factor of a tensor algebra \(\Lambda=T_A(V)/(V^{\otimes n})\) and two ideals \(I\) and \(J\) of \(\Lambda\) that are included in \(\text{rad}^2(\Lambda)\) any \(K\)-isomorphism between \(\Lambda/I\) and \(\Lambda/J\) can be lifted to an automorphism of \(\Lambda\). As a consequence any element \(f\in{\mathcal Q}_\Lambda\) with \(\text{Ann}_\Lambda(f)\subseteq\text{rad}^2(\Lambda)\) is an element of some subset \({\mathcal Q}_\Lambda(\sigma)\) for some automorphism \(\sigma\). The union \({\mathcal Q}_\Lambda^\#\) of all subsets of the form \({\mathcal Q}_\Lambda^\#(\sigma)=\{f\in{\mathcal Q}_\Lambda(\sigma)\mid\text{Ann}_\Lambda(f)\subseteq\text{rad}^2(\Lambda)\}\) is a \({\mathcal G}_\Lambda\)-space and the orbit space \({\mathcal Q}_\Lambda^\#/{\mathcal G}_\Lambda\) is in bijection with the set \({\mathcal A}_n(A,V)\) of all representatives of \(K\)-algebra isomorphism classes of Frobenius algebras \(\Gamma\) satisfying (1) \(\Gamma/\text{rad}(\Gamma)\simeq A\), (2) \(\text{rad}(\Gamma)/\text{rad}^2(\Gamma)\simeq V\), (3) \(\text{rad}^n(\Gamma)=0\).