On the representation dimension of smash products

Abstract

Let $A$ be a finite dimensional $G$-graded algebra with $G$ a finite group, and $A\# k[G]^{\ast}$ be the smash product of $A$ with the group $G$. Our results can be stated as follows: (1) If $A$ is a self-injective algebra and separably graded, then the dimensions of triangulated categories $\underline{\rm mod}A$ and $\underline{\rm mod}A\# k[G]^{\ast}$ are equal. In particular, we obtain that the representation dimension of $A\# k[G]^{\ast}$ is at least the dimension of triangulated category $\underline{\rm mod}A$ plus 2; (2) Generally, if $A$ is a $k$-algebra and separably graded, then the Oppermann dimensions of $A$ and $A\# k[G]^{\ast}$ are equal. In particular, we obtain that the representation dimension of $A\# k[G]^{\ast}$ is at least the Oppermann dimension of $A$ plus 2. In the end, we give two examples to illustrate our results.