Superalgebras with superinvolution

The purpose of this thesis is to present in the setting of superalgebras with superinvolution some of the most interesting and challenging problems of combinatorial PI-theory (the theory of polynomial identities), which have already been addressed in the field of associative algebras or of algebras with involution. More precisely, I shall characterize the varieties of superalgebras with superinvolution of polynomial growth and along the way I shall classify the subvarieties of the varieties of almost polynomial growth. Finally I shall find standard identities of minimal degree in the setting of matrix superalgebras with superinvolution and in this way I shall show that the Amitsur-Levitzki theorem can be improved by considering only certain kinds of matrices.