Some characterizations of fundamental graded algebras

One of the basic notions in the theory of varieties of algebras in characteristic zero developed by Kemer [20] was that of fundamental algebras. They are used as a main tool in the solution of Specht’s Problem. The aim of this paper is to extend this concept to algebras with a G-graded structure, where G is a finite group, and to develop the corresponding theory. Further- more, we explore the connection between fundamental G-graded algebras and generators of affine varieties of G-graded PI algebras which are minimal with respect to their G-graded exponent. In some important cases, we provide necessary and sufficient conditions so that subalgebras of these generators are fundamental. Finally, for abelian groups, we give a characterization in terms of the representation theory of the group G ≀ Sn.