Characterization of Nilpotent Lie Algebras Pair by Their Schur Multipliers

Abstract

Let (N, L) be a pair of finite dimensional nilpotent Lie algebras, in which N is an ideal in L and ℳ(N, L) is the Schur multiplier of the pair (N, L). In 2011, the second author and colleagues proved that if N admits a complement K say, in L with dim N = n and dim K = m, then , where t(N, L) is a non-negative integer. In the present article, we characterize the pairs (N, L) for which t(N, L) = 0, 1, 2, 3, and 4. Also, in a special case when L is a filiform Lie algebra, we determine all pairs (N, L), when t(N, L) = 0, 1,…, 10.

Key Words:

• Filiform Lie algebra
• Pair of Lie algebras
• Schur multiplier

2010 Mathematics Subject Classification:

• Primary 17B30, 17B60
• Secondary 17B99

ACKNOWLEDGMENT

The authors would like to thank the referee for the helpful suggestions, which made the article more readable.

Notes

Communicated by S. Sehgal.