Nilpotent Lie algebras having the Schur multiplier of maximum dimension

Abstract: Let $ L $ be an $ n $-dimensional nilpotent Lie algebra of nilpotency class $ c $ with the derived subalgebra of dimension $ m $. Recently, Rai proved that the dimension of Schur multiplier of $ L $ is bounded by $ \frac{1}{2}(n-m-1)(n+m)-\sum\limits_{i=2}^ {min\lbrace n-m,c\rbrace} n-m-i $. In this paper, we obtain the structure of all nilpotent Lie algebras that attain this bound.