Fundamental structures in PI theory

Fundamental superalgebras are finite-dimensional superalgebras over an algebraically closed field of characteristic zero, defined using certain multialternating graded polynomials. These superalgebras play a pivotal role in Kemer’s Representability Theorem. In this work, we extend the concept to algebras graded by finite groups and to algebras with involution. We provide examples in both settings, focusing on generators of affine varieties that are minimal with respect to their graded-exponent and ∗-exponent, as well as on specific subalgebras of these structures. Key results include a characterization of fundamental algebras, graded by abelian groups and with involution, via the representation theory of the wreath product of an appropriate finite group and the symmetric group. We also present a characterization of fundamental ∗-algebras and superalgebras whose radical is 1-codimensional. Along the way, we provide new results regarding fundamental algebras in the classical framework.