# Twist invariants of graded algebras

## TL;DR

This paper introduces two new invariants for graded algebras that remain unchanged under certain twists, helping distinguish quantum algebras based on their parameterization and providing tools for algebra classification.

## Contribution

The paper defines two twist invariants for graded algebras, analyzes their stability, and offers methods for computing them in various algebra families, advancing algebra classification techniques.

## Key findings

- Invariants distinguish 'truly multiparameter' from 'uniparameter' quantum algebras.
- Invariants are stable under polynomial extension.
- Methods for computing invariants are applicable to quantum nilpotent and cluster algebras.

## Abstract

We define two invariants for (semiprime right Goldie) algebras, one for algebras graded by arbitrary abelian groups, which is unchanged under twists by $2$-cocycles on the grading group, and one for $\mathbb Z$-graded or $\mathbb Z_{\ge 0}$-filtered algebras. The first invariant distinguishes quantum algebras which are "truly multiparameter" apart from ones that are "essentially uniparameter", meaning cocycle twists of uniparameter algebras. We prove that both invariants are stable under adjunction of polynomial variables. Methods for computing these invariants for large families of algebras are given, including quantum nilpotent algebras and algebras admitting one quantum cluster, and applications to non-isomorphism theorems are obtained.