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

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 circle times L) <= (n - m)(n - 1) + 2. Furthermore for m = 1, the explicit structure of L is given when the equality holds.