Decomposition of the nonabelian tensor product of Lie algebras via the diagonal ideal

TL;DR

This paper presents a splitting theorem for the nonabelian tensor product of Lie algebras, relating it to the diagonal ideal and exterior product, with implications for structure analysis and homotopy theory.

Contribution

It introduces a new splitting theorem for the nonabelian tensor product of Lie algebras involving the diagonal ideal and exterior product, extending previous results and exploring free product behavior.

Findings

The size of the diagonal ideal influences the structure of the tensor product.

A splitting theorem is proved for the tensor product in terms of the diagonal ideal and exterior product.

Connections with homotopy theory are established.

Abstract

We prove a theorem of splitting for the nonabelian tensor product $L \otimes N$ of a pair $(L,N)$ of Lie algebras $L$ and $N$ in terms of its diagonal ideal $L \square N$ and of the nonabelian exterior product $L \wedge N$. A similar circumstance was described two years ago by the second author in the special case $N=L$. The interest is due to the fact that the size of $L \square N$ influences strongly the structure of $L \otimes N$. Another question, often related to the structure of $L \otimes N$, deals with the behaviour of the operator $\square$ with respect to the formation of free products. We answer with another theorem of splitting even in this case, noting some connections with the homotopy theory.