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Journal of Lie Theory 21 (2011), No. 2, 491--498
Copyright Heldermann Verlag 2011

The Commutator Subalgebra and Schur Multiplier of a Pair of Nilpotent Lie Algebras

Farshid Saeedi
Dept. of Mathematics, Islamic Azad University, Mashhad-Branch, Iran

Ali Reza Salemkar
Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

Behrouz Edalatzadeh
Dept. of Mathematics, Faculty of Science, Razi University, Kermanshah, Iran


Let $(L,N)$ be a pair of finite dimensional nilpotent Lie algebras, in which $N$ is an ideal in $L$. In the present article, we prove that if the factor Lie algebras $L/N$ and $N/Z(L,N)$ are of dimensions $m$ and $n$, respectively, then the commutator subalgebra $[L,N]$ is of dimension at most ${1\over2}n(n+2m-1)$, and also determine when ${\rm dim}([L,N]) = {1\over2}n(n+2m-1)$. In addition, we introduce the notion of the Schur multiplier ${\cal M}(L,N)$ of an arbitrary pair $(L,N)$ of Lie algebras, and show that if $N$ admits a complement $K$ in $L$ with ${\rm dim}(N)=n$ and ${\rm dim}(K)=m$, then the dimension of ${\cal M}(L,N)$ is bounded above by ${1\over2}n(n+2m-1)$. In this case, we characterize the pairs $(L,N)$ for which ${\rm dim}({\cal M}(L,N))$ is either ${1\over2}n(n+2m-1)$ or ${1\over2}n(n+2m-1)-1$.

Keywords: Lie algebra, Schur multiplier, cover.

MSC: 17B30, 17B60, 17B99

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