Invariant properties of modules under smash products from finite dimensional algebras

Abstract

We give the relationship between indecomposable modules over the finite dimensional $ k $-algebra $ A $ and the smash product $ A\sharp G $ respectively, where $ G $ is a finite abelian group satisfying $ G\subseteq Aut(A) $, and $ k $ is an algebraically closed field with the characteristic not dividing the order of $ G $. More precisely, we construct all indecomposable $ A\sharp G $-modules from indecomposable $ A $-modules and prove that an $ A\sharp G $-module is indecomposable if and only if it is an indecomposable $ G $-stable module over $ A $. Besides, we give the relationship between simple, projective and injective modules in $ modA $ and those in $ modA\sharp G $.

Keywords

• finite dimensional algebra
• smash product
• indecomposable module
• g-stable module
• stable category
• abelian