Some notes on the schur multiplier of a pair of lie algebras

摘要

In this paper, we review some properties of the non-abelian exterior product and using them provide special properties of the Schur multiplier of a pair of Lie algebras. By constructing a cover for a pair of Lie algebras, we give a necessary and sufficient condition for the capability of a pair of Lie algebras. Also, for a perfect pair of Lie algebras we construct a cover which is a universal extension. Finally, we give some bounds for the dimension of the Schur multiplier of a pair of Lie algebras.