On dimension of Schur multiplier of nilpotent Lie algebras II

Abstract

Let [Formula: see text] be a non-abelian nilpotent Lie algebra of dimension [Formula: see text] and put [Formula: see text], where [Formula: see text] denotes the Schur multiplier of [Formula: see text]. Niroomand and Russo in 2011 proved that [Formula: see text] and that [Formula: see text] if and only if [Formula: see text], in which [Formula: see text] is the Heisenberg algebra of dimension [Formula: see text] and [Formula: see text] is the abelian [Formula: see text]-dimensional Lie algebra. In the same year, they also classified all nilpotent Lie algebras [Formula: see text] satisfying [Formula: see text] or [Formula: see text]. In this paper, we obtain all nilpotent Lie algebras [Formula: see text] provided that [Formula: see text].

Characterizing nilpotent Lie algebras rely on the dimension of their $2$-nilpotent multipliers

“…There are several papers devoted to investigation of the structure of an n-dimensional nilpotent non-abelian Lie algebra L rely on s(L). The structure of all nilpotent non-abelian Lie algebras L is obtain when s(L) = 0, 1, 2, 3 in [7,8,12]. These results not only characterize a nilpotent Lie algebra in terms of s(L) but also they can help to shorten the processes of finding the structure of a nilpotent Lie algebra L in terms of t(L) = 1 2 n(n − 1) − dim M(L)…”

On characterizing nilpotent Lie algebras by their multiplier, $$s(L)=4$$

“…. , 5 are given in [12,14,16,21,22]. This classification simplify the problem of determining a Lie algebra in terms of t(L) and recently are used to answer the same question for an n-Lie super algebra and Leibniz n-algebras in [17,18].…”

Multipliers, Covers and Stem Extensions for Lie Superalgebras

“…The study on multipliers of Lie algebras began in 1990's (see [3,12], for example) and the theory has seen a fruitful development (see [2,5,8,9,14,17,20], for example). Among them, a typical fact analogous to the one in the group theory is that the multiplier of a finite-dimensional Lie algebra L is isomorphic to the second cohomology group of L with coefficients in the 1-dimensional trivial module (see [1], for example).…”