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 Aut Alg (A) of A is isomorphic to the group U G(a(A) o ) of all invertible group-like elements of the finite dual a(A) 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 Hom BiAlg a(A), k[G] / ≈ of all bialgebra maps a(A) → k[G], via the equivalence relation implemented by the conjugation with an invertible group-like element of a(A) o . 2010 Mathematics Subject Classification. 16W20, 16W50, 16T10. Key words and phrases. automorphisms group of associative algebras, classifications of gradings.