A characterization of algebras generated by idempotents
This paper is devoted to finding a method to determine whether an algebra is generated by idempotents. The main result of the paper is as follow. Theorem. Let \(A\) be a finite dimensional \(F\)-algebra. Then \(A\) is generated by idempotents if and only if \(\mathrm{dim}_FV>\dim_F\mathrm{End}_A(V)\) or \(\dim_F\mathrm{End}_A(V)=1\) for each irreducible \(A\)-module \(V\) and \(\mathrm{Ext}_A^1(S,S)=0\) for all \(1\)-dimensional \(A\)-modules \(S\). As consequences of the theorem, the authors also point out that many well-known algebras are generated by idempotents. For examples, let \(A\) be a finite dimensional splitting \(F\)-algebra. If \(A\) has a finite global dimension, then \(A\) is generated by idempotents. Moreover, if \(A\) is a quasi-hereditary algebra over a splitting field \(F\), then \(A\) is generated by idempotents. Particularly, \(q\)-Schur algebras and symplectic \(q\)-Schur algebras are generated by idempotents.
scientific article; zbMATH DE number 7357212
scientific article