Central cocharacters of the subvarieties of varieties of superalgebras with almost polynomial growth

Abstract: In recent years, the study of the $T$-space of central polynomials of an algebra $A$ has become an object of great interest in the PI-theory. Such interest has been extended to the context of algebras with additional structures. The main goal of this paper is to present information about the central graded codimensions and the central graded cocharacters of the varieties of superalgebras $\mathrm{var}^{gr}(G)$, $\mathrm{var}^{gr}(UT_2)$, $\mathrm{var}^{gr}(G^{gr})$, $\mathrm{var}^{gr}(UT^{gr}_2)$ and $\mathrm{var}^{gr}(D^{gr})$, which are the only supervarieties with almost polynomial growth of the graded codimensions. Also we establish the generators of the space of central polynomials, determine the central codimensions and explicitly give the decomposition of the central graded cocharacters of each minimal subvariety of such supervarieties.