Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions

Abstract

Let A be a superalgebra with graded involution or superinvolution * and let c * n (A), n = 1, 2, . . . , be its sequence of * -codimensions. In case A is finite dimensional, in [6,15] it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of Z 2 and a 4-dimensional subalgebra of the 4×4 upper-triangular matrices with suitable graded involutions or superinvolutions.In this paper we study the general case of * -superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of * -superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in [18].

Minimal varieties of PI-superalgebras with graded involution

“…, A m ) and consider a subvariety U * Z 2 ⊆ V * Z 2 (A) such that exp * Z 2 (V * Z 2 (A)) = exp * Z 2 (U * Z 2 ). Since V * Z 2 (A) satisfies a Capelli identity of some rank, according to Theorem 5.2 of [8] we conclude that U * ). This means that, eventually replacing (k 1 , .…”

“…generated by a finitely generated * -superalgebra satisfying an ordinary polynomial identity. In this case, according of Theorem 5.2 of [8], any subvariety of such a variety has finite basic rank as well. Our aim is to characterize those which are minimal with respect to their * -graded exponent.…”

“…According to the deductions of Section 2, we can suppose that F is algebraically closed. Since A is a finitely generated PI * -superalgebra, by virtue of Theorem 5.3 of [8] there exists a finite-dimensional * -superalgebra B such that Id * Z 2 (A) = Id * Z 2 (B). Eventually replacing A with B, invoking Lemma 4.2, for our aims we can also assume that A is triangular and let…”

“…In the present paper we deal with * -superalgebras, namely superalgebras endowed with a graded involution, and the growth of their * -graded codimensions, as extensively made by several authors in the last few years (see, for instance, [7], [8], [10], [16] and [19]). Since a * -superalgebra can be viewed as an algebra with generalized F G-action, where G = Z 2 × Z 2 acts on it by automorphisms and antiautomorphisms, the existence of the * -graded exponent has been confirmed in [15] in the finite-dimensional case.…”

Trace identities and almost polynomial growth

“…We recall here that similar descriptions for the ordinary codimensions can be found in [14,Theorem 7.2.7] where it was proved that the only two varieties of (ordinary) algebras of almost polynomial growth are the ones generated by the Grassmann algebra E and by U T 2 . In the case of algebras with involution, superinvolution, pseudoinvolution or graded by a finite group, a complete list of varieties of algebras of almost polynomial growth was exihibited in [5,6,9,10,18,19,20,29].…”