# Identities of graded simple algebras

## TL;DR

This paper investigates polynomial bounds on identities of finite dimensional graded algebras over characteristic zero fields, establishing the existence of graded PI-exponents for simple algebras under various conditions.

## Contribution

It proves polynomial bounds on graded colength and establishes the existence of graded PI-exponents for graded simple algebras, extending previous results to more general groupoid gradings.

## Key findings

- Graded colength has polynomially bounded growth.
- Existence of graded PI-exponent for graded simple algebras with commutative semigroup grading.
- Existence of graded PI-exponent for non-graded simple algebras without restrictions on grading.

## Abstract

We study identities of finite dimensional algebras over a field of characteristic zero, graded by an arbitrary groupoid $\Gamma$. First we prove that its graded colength has a polynomially bounded growth. For any graded simple algebra $A$ we prove the existence of the graded PI-exponent, provided that $\Gamma$ is a commutative semigroup. If $A$ is simple in a non-graded sense the existence of the graded PI-exponent is proved without any restrictions on $\Gamma$.