A generalization of Cayley-Hamilton algebras and an introduction to their geometries

Let A and B be graded algebras in the same variety of trace algebras, such that A is a finite-dimensional, central simple power associative algebra (in the ordinary sense). Over a field K of characteristic zero, we study sufficient conditions that ensure B to be a graded subalgebra of A. More precisely, we prove, under additional hypotheses, that there is a graded and trace-preserving embedding from B to A over some associative and commutative K-algebra C if and only if B satisfies all G-trace identities of A over K. As a consequence of these results, we give a geometric interpretation of our main theorem under the context of graded algebras, and we apply them beyond the Cayley-Hamilton algebras presented in [24, 29]. Such results open a wide range of opportunities to study geometry in Jordan and alternative algebras (with trivial grading).