The automorphisms group and the classification of gradings of finite dimensional associative algebras

TL;DR

This paper establishes a connection between the automorphism group of a finite-dimensional algebra and its quantum symmetry semigroup, providing a classification of all possible group gradings on the algebra.

Contribution

It explicitly describes the automorphism group in terms of invertible group-like elements and classifies all group gradings via bialgebra maps, linking algebra symmetries to quantum structures.

Findings

Automorphism group is isomorphic to the group of invertible group-like elements of the dual quantum symmetry semigroup.

All G-gradings on the algebra are classified by bialgebra maps from the quantum symmetry semigroup to the group algebra.

The set of isomorphism classes of G-gradings corresponds to a quotient of bialgebra maps modulo conjugation.

Abstract

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual $a (A)^{\rm o}$. For a group $G$, all $G$-gradings on $A$ are explicitly described and classified: the set of isomorphisms classes of all $G$-gradings on $A$ is in bijection with the quotient set $ {\rm Hom}_{\rm BiAlg} \, \bigl( a (A) , \, k[G] \bigl)/\approx$ of all bialgebra maps $a (A) \, \to k[G]$, via the equivalence relation implemented by the conjugation with an invertible group-like element of $a (A)^{\rm o}$.