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

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

2010-02-12

arxiv.org/abs/1002.2563v2

Mathematics

Rings and Algebras

Paper in press in Linear Multilinear Algebra

Scientific paper

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.

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