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

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 ${\mathrm{Aut}}_{\mathrm{Alg}}(A)$ of $A$ is isomorphic to the group $U(G(a(A{)}^{\mathrm{o}}))$ of all invertible group-like elements of the finite dual $a(A{)}^{\mathrm{o}}$. 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 ${\mathrm{Hom}}_{\mathrm{BiAlg}}(a(A),k[G])/\approx$ of all bialgebra maps $a(A)\to k[G]$, via the equivalence relation implemented by the conjugation with an invertible group-like element of $a(A{)}^{\mathrm{o}}$.

Continues (associative algebra version of) arXiv:2006.00711

References in corpus (2)

• A new invariant for finite dimensional Leibniz/Lie algebras
• The Finite Duals of Affine Prime Regular Hopf Algebras of GK-Dimension One

Cited by in corpus (3)

• Universal Coacting Hopf algebra of a Finite dimensional Lie-Yamaguti algebra
• Universal Coacting Hopf Algebra of a finite-dimensional Algebra over an Operad
• Automorphism group of Suzuki's Hopf algebra