The capability and certain functors of some nilpotent Lie algebras of class two

TL;DR

This paper extends the calculation of Schur multipliers and related functors to certain nilpotent Lie algebras of class two with specific ranks, providing new insights into their algebraic structure.

Contribution

The authors generalize previous results to a broader class of generalized Heisenberg Lie algebras with specific ranks, exploring their functors and structural properties.

Findings

Computed Schur multipliers for new rank ranges

Determined non-abelian tensor and exterior squares for these algebras

Established consequences for nilpotent Lie algebras of class two with given dimensions

Abstract

Recently, the authors obtained the Schur multiplier, the non-abelian tensor square and the non-abelian exterior square of $d$-generator generalized Heisenberg Lie algebras of rank $ \frac{1}{2}d(d-1).$ Here, we intend to obtain the same results for $d$-generator generalized Heisenberg Lie algebras of rank $ t$ when $ \frac{1}{2}d(d-1)-3 \leq t\leq \frac{1}{2}d(d-1)-1.$ Then, as a result, we give similar consequences for a nilpotent Lie algebra $ L $ of class two when $ \dim (L/Z(L))=d,$ $ \dim L^2=t $ such that $ \frac{1}{2}d(d-1)-3 \leq t\leq \frac{1}{2}d(d-1)-1.$