On the identities of 3 × 3 matrices with orthosymplectic superinvolution. Algebras with superautomorphism and codimension growth

This thesis presents results concerning PI-algebras endowed with distinct additional structures.First, we consider M_{1,2}(F), the algebra of 3 ×3 matrices with orthosymplectic superinvolution ∗ over a field F of characteristic zero. We study the ∗-identities of this algebra through the representation theory of the group Hn = (Z2 ×Z2)∼Sn.To this end, we decompose the space of multilinear ∗-identities of degree n into the sum of irreducibles under the action of Hn and we study the irreducible characters appearing in this decomposition with non-zero multiplicity. Finally, by using the representation theory of the general linear group, we determine all the ∗-polynomial identities of M_{1,2}(F) up to degree 3.Next, we focus on superalgebras endowed with a superautomorphism of order ≤2. We characterize those superalgebras whose cocharacter multiplicities are bounded by a constant. Furthermore, we determine a characterization of the superalgebras with superautomorphism with polynomial growth of the codimensions and we give a classification of the subvarieties of the varieties of almost polynomial growth. Lastly, we characterize the superalgebras with superautomorphism with linear codimension growth.