On the tensor square of non-abelian nilpotent finite-dimensional Lie algebras

Abstract

For every finite p-group G of order p n with derived subgroup of order p m , Rocco [N.R. Rocco, On a construction related to the nonabelian tensor square of a group, Bol. Soc. Brasil. Mat. 1 (1991), pp. 63–79] proved that the order of tensor square of G is at most p n(n−m). This upper bound has been improved recently by the author [P. Niroomand, On the order of tensor square of non abelian prime power groups (submitted)]. The aim of this article is to obtain a similar result for a non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any given n-dimensional non-abelian nilpotent Lie algebra L with derived subalgebra of dimension m we have dim(L ⊗ L) ≤ (n − m)(n − 1) + 2. Furthermore for m = 1, the explicit structure of L is given when the equality holds.