Superálgebras com involução graduada: classificação das variedades minimais de crescimento quadrático

Abstract

Let V be a variety of superalgebras with graded involution and let $\{\cgri(v)\}_{n\geq 1}$ be its sequence of *-graded codimensions. We say that V has polynomial growth $n^k$ if asymptotically $\cgri(V)\approx an^k$, for some $a\ne 0$. Furthermore, V is minimal of polynomial growth $n^k$ if $\cgri(V)$ grows as $n^k$ and any proper subvariety of V has polynomial growth $n^t$, with $t<k$. In this thesis we present the classification of minimal varieties of superalgebras with graded involution with quadratic growth, by giving a complete list of 36 finite dimensional superalgebras with graded involution which generate, up to equivalence, the only minimal varieties of quadratic growth. The 36 superalgebras with graded involution presented here form the smallest list of algebras that should be excluded from a variety V in order to conclude that V has at most linear growth. We emphasize that among these algebras, 16 are presented in an unprecedented way in this work.