Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions

Let A be a superalgebra with graded involution or superinvolution and let cn</mml:msubsup>(A), n =1,2,..., be its sequence of -codimensions. In case A is finite dimensional, in Giambruno et al. (Algebr. Represent. Theory 19(3), 599-611 2016, Linear Multilinear Algebra 64(3), 484-501 2016) it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of <mml:msub>Z2 and a 4-dimensional subalgebra of the 4 x 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of -superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of -superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in Ioppolo and La Mattina (J. Algebra 472, 519-545 2017).