The index of Lie poset 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 discrete Morse theory argument establishes that the simplicial realizations of such posets are 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, such a Lie poset algebra is absolutely rigid and cannot be deformed. We also provide matrix representations for Lie poset algebras in the other classical types.