On the representation dimension of smash products

TL;DR

This paper investigates how the representation dimension of a finite-dimensional G-graded algebra's smash product relates to the original algebra, establishing equalities and bounds for various dimensions under certain conditions.

Contribution

It provides new results linking the representation and triangulated category dimensions of a G-graded algebra and its smash product, especially under separable grading.

Findings

Dimensions of triangulated categories are equal for the algebra and its smash product.

Representation dimension of the smash product is at least the original's plus two.

Examples illustrate the theoretical results.

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.