The Automorphisms Group and the Classification of Gradings of Finite Dimensional Associative Algebras

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 is isomorphic to \(U \bigl ( G(a (A)^{\mathrm{o}} ) )\) all invertible group-like elements finite dual \(a (A)^{\mathrm{o}}\). For G, G-gradings on are explicitly described classified: set isomorphisms classes in bijection with quotient \( \mathrm{Hom}_{\mathrm{BiAlg}} \, (A) , k[G] )/\approx \) bialgebra maps \rightarrow k[G]\), via equivalence relation implemented by conjugation element (A)^\mathrm{o}\).