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

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.

The published version contains some improvements

References in corpus (1)

• Estimations of the low dimensional homology of Lie algebras with large abelian ideals

Cited by in corpus (2)

• Capability of nilpotent Lie algebras of small dimension
• The structure of nilpotent Lie algebras of class two with derived subalgebra of dimension two