Semisimplicity of Locally Finite Graded R#H-Modules

Abstract

Let k be a field, Γ an abelian group with a bicharacter, R a colour algebra over k (i.e., a Γ-graded associative k-algebra with identity), H a Hopf colour k-algebra acting on R in such a way that R is a graded H-module algebra and the associated smash product R#H is a colour algebra. The aim of this paper is to study the semisimplicity of the category of H-locally finite Γ-graded R#H-modules. From our main result we deduce that if H is finite-dimensional and R is left graded-noetherian and graded-semisimple, then the colour algebra R#H is graded-semisimple if either H is graded-semisimple or if H is colour-cocommutative and R is colour-commutative and projective in the category of graded R#H-modules.