# Semisimplicity of affine cellular algebras

## TL;DR

This paper characterizes when affine cellular algebras are semisimple, linking algebraic properties to geometric conditions and providing criteria for separability and Jacobson semisimplicity.

## Contribution

It establishes necessary and sufficient conditions for the semisimplicity of affine cellular algebras, including geometric and bilinear form criteria, and explores separability and Jacobson semisimplicity.

## Key findings

- Affine cellular algebra is semisimple iff associated scheme is reduced and 0-dimensional.
- Semisimplicity is equivalent to all bilinear forms being isomorphisms.
- Over a perfect field, semisimplicity coincides with separability.

## Abstract

In this note, we prove that an affine cellular algebra $A$ is semisimple if and only if the scheme associated to $A$ is reduced and 0-dimensional, and the bilinear forms with respect to all layers of $A$ are isomorphisms. Moreover, if the ground ring is a perfect field, then $A$ is semisimple if and only if it is separable. We also give a sufficient condition for an affine cellular algebra being Jacobson semisimple.