# Separable equivalence, complexity and representation type

## TL;DR

This paper extends the concept of separable equivalence to additive categories, demonstrating its preservation of representation type and providing examples of algebras with the same complexity but not separably equivalent.

## Contribution

It generalizes separable equivalence to additive categories and shows its implications for representation type and algebra classification.

## Key findings

- Separable equivalence preserves the representation type of an algebra.
- Certain small cyclic group algebras are not separably equivalent despite having the same complexity.
- The generalization allows constructing multiple related equivalences from an initial one.

## Abstract

We generalise the notion of separable equivalence, originally presented by Linckelmann (2011), to an equivalence relation on additive categories. We use this generalisation to show that from an initial equivalence between two algebras we may build equivalences between many related categories. We also show that separable equivalence preserves the representation type of an algebra. This generalises Linckelmann's result, where he showed this in the case of symmetric algebras. We use these theorems to show that the group algebras of several small cyclic groups cannot be separably equivalent. This gives several examples of algebras that have the same complexity but are not separably equivalent.