Finite group actions on dg categories and Hochschild homology

TL;DR

This paper establishes a decomposition of Hochschild homology for dg categories with finite group actions, linking it to categorical actions of representation categories and the representation ring, enhancing understanding of equivariant dg categories.

Contribution

It introduces a new decomposition of Hochschild homology for equivariant dg categories under finite group actions, connecting it to categorical and algebraic structures.

Findings

Decomposition of Hochschild homology for equivariant dg categories.

Relation between the decomposition and categorical actions of Rep(G).

Action of the representation ring R_C(G) on Hochschild homology.

Abstract

We prove a decomposition of the Hochschild homology groups of the equivariant dg category $\mathscr{C}^G$ associated to a small dg category $\mathscr{C}$ with direct sums on which a finite group $G$ acts. When the ground field is $\mathbb{C}$ this decomposition is related to a categorical action of $\text{Rep}(G)$ on $\mathscr{C}^G$ and the resulting action of the representation ring $R_\mathbb{C}(G)$ on $HH_\bullet(\mathscr{C}^G)$.