Polynomial identities on superalgebras and exponential growth

Abstract

Let A be a finitely generated superalgebra over a field F of characteristic 0. To the graded polynomial identities of A one associates a numerical sequence { c n sup( A)} n⩾1 called the sequence of graded codimensions of A. In case A satisfies an ordinary polynomial identity, such sequence is exponentially bounded and we capture its exponential growth by proving that for any such algebra lim n→∞ c n sup (A) n exists and is a non-negative integer; we denote such integer by supexp( A) and we give an effective way for computing it. As an application, we construct eight superalgebras A i , i=1,…,8, characterizing the identities of any finitely generated superalgebra A with supexp( A)>2 in the following way: supexp( A)>2 if and only if Id sup( A)⊆ Id sup( A i ) for some i∈{1,…,8}, where Id sup( B) is the ideal of graded identities of the algebra B. We also compare the superexponent and the exponent (see A. Giambruno, M. Zaicev, Adv. Math. 140 (1998) 145–155) of any finitely generated superalgebra.