The 2-nilpotent multiplier of n-Lie algebras and its applications
Farshid Saeedi, Seyedeh Nafiseh Akbarossadat
TL;DR
This work analyzes the $2$-nilpotent multiplier $\mathcal{M}^{(2)}(L)$ of finite-dimensional $n$-Lie algebras and its role in $2$-capability. It derives a precise decomposition for $\mathcal{M}^{(2)}(L\oplus M)$, computes dimensions for abelian and Heisenberg families, and provides a general dimension formula for nilpotent $n$-Lie algebras based on the dimension of the derived subalgebra. The authors establish $2$-capability criteria via $Z_2^*(L)$ and show that among Heisenberg algebras only $H(1)$ is $2$-capable, highlighting the interplay between central extensions and higher-arity Lie algebra structure. These results advance understanding of higher-order multipliers, central extensions, and capability in $n$-Lie algebras with potential implications for their cohomology and classification.
Abstract
In this paper, we first recall the concept of c-nilpotent multiplier and c-capability of n-Lie algebras and also, recall the formula for calculating the number of basic commutators in n-Lie algebras. Then we give the structure of 2-nilpotent multiplier of the direct sum of two n-Lie algebras. Next, we calculate the dimension of 2-nilpotent multiplier of every abelian n-Lie algebras and Heisenberg n-Lie algebras H(n,m). Then we give a dimension of 2-nilpotent multiplier of any nilpotent n-Lie algebras of class 2 by using the number of basic commutators.