# The G-centre and gradable derived equivalences

## TL;DR

This paper introduces the G-centre, a generalization of the algebra center for G-graded algebras, and demonstrates its invariance under gradable derived equivalences, with applications to superalgebras.

## Contribution

It defines the G-centre for G-graded algebras, explores its properties, and shows its invariance under certain derived equivalences, connecting to superalgebra theory.

## Key findings

- G-centre controls endomorphism categories of grading shifts
- G-centre is preserved under gradable derived equivalences
- Applications to derived equivalences of superalgebras

## Abstract

We propose a generalisation for the notion of the centre of an algebra in the setup of algebras graded by an arbitrary abelian group G. Our generalisation, which we call the G-centre, is designed to control the endomorphism category of the grading shift functors. We show that the G-centre is preserved by gradable derived equivalences given by tilting modules. We also discuss links with existing notions in superalgebra theory and apply our results to derived equivalences of superalgebras.