Lie algebras and coverings
Let \(\Lambda\) be an associative unitary finite-dimensional \(\mathbb{C}\)-algebra which is representation finite. This means that the number of isomorphism classes of indecomposable finite-dimensional \(\Lambda\)-left modules is finite. Let us fix a set \(\mathcal F\) of representatives for these isomorphism classes. We showed [\textit{C. Riedtmann}, J. Algebra 170, No. 2, 526-546 (1994; Zbl 0841.16018)] that the free \(\mathbb{Z}\)-module \(L(\Lambda)=\bigoplus_{A\in{\mathcal F}}\mathbb{Z} v_A\) generated by the symbols \(\{v_A:A\in{\mathcal F}\}\) can be made into a \(\mathbb{Z}\)-Lie algebra. This is the complex version of \textit{C. M. Ringel}'s construction of Lie algebras via Hall algebras over finite fields [Banach Center Publ. 26, 433-447 (1990; Zbl 0778.16004)]. The construction of \(L(\Lambda)\) carries over easily to the case where \(\Lambda\) is a locally representation finite \(\mathbb{C}\)-category. Theorem. Let \(\Lambda\) be a locally representation finite \(\mathbb{C}\)-category with universal cover \(\widetilde\Lambda\) and fundamental group \(G\). Then \(L(\Lambda)\) is isomorphic to \(L(\widetilde\Lambda)/G\).
indecomposable finite-dimensional left modules
locally representation finite categories
scientific article; zbMATH DE number 720098