Trace identities and almost polynomial growth

Abstract

In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: D 2 , the algebra of 2 × 2 diagonal matrices and C 2 , the algebra of 2 × 2 matrices generated by e 11 + e 22 and e 12. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties with trace of polynomial growth generated by a finite dimensional algebra. As a consequence, we see that the growth of a variety with trace is either polynomial or exponential.

Matrix algebras with degenerate traces and trace identities

“…As we have seen in Theorem 17, the number of monomials in (3) and ( 4) is 2 n and S(n + 1, 3), respectively. In order to complete the proof we need to show that in (5) there are exactly nS(n, 3) elements. Notice that we can choose the variable x s in n distinct ways.…”

“…Definition 23. A variety V of algebras with trace is minimal of polynomial growth if c tr n (V) ≈ qn k , for some k ≥ 1, q > 0, and for every proper subvariety U V generated by a unitary finite dimensional algebra, we have that c tr n (U) ≈ q ′ n t , with t < k. The following result (see [5,Theorem 30]) gives a characterization of the varieties of polynomial codimension growth.…”

“…Then we prove that the trace identities of D n+1 endowed with a degenerate trace, are very closely related to those of D n with the trace which is in a sense the restriction of the trace of D n+1 to D n . As a consequence, starting from the trace identities of D 2 , determined in [5,6], we find the generators of the trace T -ideal of the identities of D 3 endowed with all possible degenerate traces. Along the way we also compute the trace codimension sequence of these algebras.…”

“…The last part of the paper is devoted to the study of varieties of polynomial growth. We completely classify the subvarieties generated by unitary finite dimensional algebras of the varieties of algebras of almost polynomial growth ( [5]). Recall that these are varieties whose codimensions grow exponentially but for each proper subvariety they grow polynomially.…”

“…In [5] we characterized the varieties generated by finite dimensional algebras whose trace codimensions are of polynomial growth .…”