Some properties on the tensor square of Lie algebras

TL;DR

This paper investigates the structure of the tensor square of Lie algebras, providing new decompositions and isomorphisms, especially for those with finite-dimensional quotients, and offers a free presentation for related algebraic constructs.

Contribution

It extends previous results by establishing new isomorphisms and decompositions for the tensor square of Lie algebras with finite-dimensional quotients, and constructs a free presentation for the wedge product.

Findings

Proves $L ext{ extperiodcentered}L ext{ extperiodcentered} ext{ is isomorphic to } L ext{ extperiodcentered} ext{L} ext{ extperiodcentered} ext{ and } L ext{ extperiodcentered} ext{wedge} L$.

Shows $L ext{ extperiodcentered} ext{wedge} L$ is isomorphic to the derived subalgebra of a cover of $L$.

Provides a free presentation for the wedge product of Lie algebras.

Abstract

In the present paper we extend and improve the results of \cite{bl, br} for the tensor square of Lie algebras. More precisely, for any Lie algebra $L$ with $L/L^2$ of finite dimension, we prove $L\otimes L\cong L\square L\oplus L\wedge L$ and $Z^{\wedge}(L)\cap L^2=Z^{\otimes}(L)$. Moreover, we show that $L\wedge L$ is isomorphic to derived subalgebra of a cover of $L$, and finally we give a free presentation for it.