On codimension growth of finite-dimensional Lie superalgebras

Abstract

Let A be a finite-dimensional simple algebra over a field of characteristic zero and cn(A), n=1, 2, …, its sequence of codimensions. Here we prove that e x p ( A ) = l i m n → ∞ c n ( A ) n , the PI-exponent of A, exists and is bounded from above by dim A. It is well known that, for associative or Lie or Jordan algebras, the equality exp (A)=dim A holds, provided that the main field is algebraically closed. Since simple Lie superalgebras are simple in a non-graded sense, their PI-exponent exists and here we prove that for the infinite family of Lie superalgebras of type b(t), t⩾3, the PI-exponent is strictly less than the dimension. Finally, we exhibit a seven-dimensional Lie superalgebra whose PI-exponent is strictly between 6 and 7.