Some properties on the tensor square 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.

To appear in J. Algebra Appl. with the small revision

References in corpus (3)

• A note on the Schur multiplier of a nilpotent Lie algebra
• Structure of nilpotent Lie algebra by its multiplier
• On the tensor square of non-abelian nilpotent finite dimensional Lie algebras

Cited by in corpus (8)

• The structure, capability and the Schur multiplier of generalized Heisenberg Lie algebras
• Some results on the Schur multiplier of nilpotent Lie algebras
• Certain functors of nilpotent Lie algebra with the derived subalgebra of dimension two
• Decomposition of the nonabelian tensor product of Lie algebras via the diagonal ideal
• Capability of nilpotent Lie algebras of small dimension
• The structure of nilpotent Lie algebras of class two with derived subalgebra of dimension two
• The capability and certain functors of some nilpotent Lie algebras of class two
• On the triple tensor product of nilpotent Lie algebras