Contact Lie poset algebras

TL;DR

This paper characterizes all height-two posets whose associated type-A Lie poset algebras are contact, showing their contractibility and absolute rigidity through combinatorial and topological methods.

Contribution

It provides a complete combinatorial classification of contact Lie poset algebras of height at most two and proves their rigidity using discrete Morse theory and cohomology.

Findings

All such posets are classified combinatorially.

Connected posets' realizations are contractible.

Corresponding Lie algebras are absolutely rigid.

Abstract

We provide a combinatorial recipe for constructing all posets of height at most two for which the corresponding type-A Lie poset algebra is contact. In the case that such posets are connected, a discrete Morse theory argument establishes that the posets' simplicial realizations are contractible. It follows from a cohomological result of Coll and Gerstenhaber on Lie semi-direct products that the corresponding contact Lie algebras are absolutely rigid.