Álgebras com estruturas adicionais de crescimento polinomial

Abstract

The classic Kemer's Theorem, established in $1979$, states that a variety of algebras $V$ has polynomial growth if, and only if, $UT_2, \mathcal{G} \notin V$. The Kemer’s caracterization was extended to algebras with additional structures by other authors. In $2001$, Giambruno and Mishchenko proved that a necessary and sufficient condition to have $V$ as a $*$-variety of polynomial growth is excluding the $*$-algebras $D_*$ and $M_*$ from $V$. In the same year, Giambruno, Mishchenko and Zaicev characterized varieties of superalgebras with polynomial growth by the exclusion of five superalgebras from the variety of superalgebras, which are: $UT_2$, $\mathcal{G}$, $UT_2^{gr}$, $\mathcal{G} ^{gr}$ and $D^{gr}$. Finally, in $2016$, Giambruno, dos Santos and Vieira proved that it is necessary and sufficient to exclude the $*$-superalgebras $D_*$, $M_*$, $D^{gr}$, $D^{gri}$ and $M^{gri}$ from a variety of $*$-superalgebras in order to have polynomial growth. The main purpose of this dissertation is to present the previous characterizations, giving proofs with updated language developed in PI-theory in the last years.