The index of Lie poset algebras

TL;DR

This paper derives formulas for the index of type-A Lie poset algebras with restricted height, characterizes Frobenius cases, and explores their rigidity and deformability, including examples of solvable, Frobenius Lie algebras.

Contribution

It provides explicit formulas for the index, a combinatorial construction for Frobenius cases, and matrix representations that identify deformable solvable Frobenius Lie algebras.

Findings

Formulas for the index of type-A Lie poset algebras.

Characterization of Frobenius Lie poset algebras of heights zero, one, and two.

Existence of deformable solvable Frobenius Lie algebras.

Abstract

We provide general closed-form formulas for the index of type-A Lie poset algebras corresponding to posets of restricted height. Furthermore, we provide a combinatorial recipe for constructing all posets corresponding to type-A Frobenius Lie poset algebras of heights zero, one, and two. A finite Morse theory argument establishes that the simplicial realization of such posets is contractible. It then follows, from a recent theorem of Coll and Gerstenhaber, that the second Lie cohomology group of the corresponding Lie poset algebra with coefficients in itself is zero. Consequently, the Lie poset algebra is absolutely rigid and cannot be deformed. We also provide matrix representations for Lie poset algebras in the other classical types. By so doing, we are able to give examples of deformable Lie algebras which are both solvable and Frobenius. This resolves a question of Gerstenhaber and Giaquinto about the existence of such algebras.