On the Homology of Nilpotent k-ary Lie Algebras

2020/06/06 by Emre Sen, Sen, Emre
Mathematics · #05E15 #17B30 #17B55 #17B56 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2006.03794

openalex publication_date 2020/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The homology groups of some two-step nilpotent Lie algebras associated to symplectic vector spaces

Related

• n-ary Lie and Associative Algebras
• The homology of Heisenberg Lie algebras over fields of characteristic\n two
• On derivations and low-dimensional (co)homology groups of pro-sovable Lie algebras associated with n₁ and n₂
• On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven
• Cohomology of some families of Lie algebras and quadratic Lie algebras