On simple modules over twisted finite category algebras. (Q2846727)

The main result of this paper is the following: Theorem. Let \(\Bbbk\) be a commutative ring and \(\mathcal C\) a finite category. Let \(\alpha\) be a \(2\)-cocycle of \(\mathcal C\) with coefficients in \(\Bbbk^*\). The map sending a simple module over the \(\alpha\)-twisted \(\Bbbk\)-linear category \(\Bbbk_\alpha\mathcal C\) to a pair \((e,\widehat eS)\), where \(e\) is an idempotent endomorphism of \(\mathcal C\), minimal with respect to \(\widehat eS\neq\{0\}\), where \(\widehat e:=\alpha(e,e)^{-1}e\), induces a bijection between the set of isomorphism classes of simple \(\Bbbk_\alpha\mathcal C\)-modules and the set of isomorphism classes of pairs \((e,T)\) consisting of an idempotent endomorphism \(e\) in \(\mathcal C\) and a simple \(\Bbbk_\alpha\mathcal C_e\)-module \(T\).NEWLINENEWLINE This theorem applies to many classes of algebras, including algebras of finite semigroups and many diagram algebras.