On the Homology of Nilpotent k-ary Lie Algebras

TL;DR

This paper explores the homology of nilpotent k-ary Lie algebras, providing explicit Betti number formulas, stability results, and verifying the toral rank conjecture for specific classes, including Heisenberg and free nilpotent cases.

Contribution

It introduces new classes of nilpotent k-ary Lie algebras, develops methods to compute their homology, and proves stability and conjecture verification results.

Findings

Explicit Betti number formulas for certain classes

Representation stability of free nilpotent k-ary Lie algebras

Verification of the toral rank conjecture for studied classes

Abstract

We introduce nilpotent k-ary Lie algebras including analogues of Heisenberg Lie algebras and free nilpotent Lie algebras. We study homology of k-ary nilpotent Lie algebras by using a modification of Chevalley-Eilenberg complex. For some classes of nilpotent k-ary Lie algebras and in particular Heisenberg k-ary Lie algebras we give explicit formulas for Betti numbers. Representation stability of free nilpotent k-ary Lie algebras is proven and lower bounds for Betti numbers are described by Schur modules. We also verify that toral rank conjecture holds for the classes we studied. Moreover, for 2-step nilpotent k-ary Lie algebras, we give a refinement of it.