# On weak equivalences of gradings

## TL;DR

This paper investigates the concept of weak equivalence of gradings on algebras, characterizing when such gradings are equivalent to finite group gradings through properties of the universal group, and provides bounds for matrix algebra gradings.

## Contribution

It introduces the notion of weak equivalence of gradings, characterizes when gradings are equivalent to finite group gradings via the universal group, and establishes bounds for matrix algebra gradings.

## Key findings

- A grading is weakly equivalent to a finite group grading iff the universal group is residually finite.
- All elementary gradings on M_n(F) are weakly equivalent to finite group gradings for n ≤ 3.
- Existence of elementary gradings on M_n(F) not weakly equivalent to finite group gradings for n ≥ 349.

## Abstract

When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. The following question arises naturally: when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group? It turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading. Namely, a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group. The same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset. We show that if $n\geqslant 349$, then on the full matrix algebra $M_n(F)$ there exists an elementary group grading that is not weakly equivalent to any grading by a finite (semi)group, and if $n\leqslant 3$, then any elementary grading on $M_n(F)$ is weakly equivalent to an elementary grading by a finite group.