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

TL;DR

This paper extends bounds on the tensor square size from finite p-groups to non-abelian nilpotent Lie algebras, providing an upper limit on their tensor square dimension and characterizing the structure when the bound is tight.

Contribution

It establishes an upper bound on the tensor square dimension of non-abelian nilpotent Lie algebras and characterizes the structure when the bound is achieved.

Findings

Upper bound of (n-m)(n-1)+2 for tensor square dimension

Explicit structure description when m=1 and equality holds

Extension of group-theoretic bounds to Lie algebra context

Abstract

For every finite $p$-group $G$ of order $p^n$ with derived subgroup of order $p^m$, Rocco in \cite{roc} proved that the order of tensor square of $G$ is at most $p^{n(n-m)}$. This upper bound has been improved recently by author in \cite{ni}. The aim of the present paper 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 $\mathrm{dim} (L\otimes L)\leq (n-m)(n-1)+2$. Furthermore for $m=1$, the explicit structure of $L$ is given when the equality holds.