(Co)Homology of Poset Lie Algebras

TL;DR

This paper explores the (co)homological properties of Lie algebras constructed from finite posets, confirming conjectures about torsion and providing explicit formulas for various algebra families using algebraic Morse theory.

Contribution

It confirms conjectures on torsion in homology of poset Lie algebras and introduces new explicit formulas for their (co)homology, connecting combinatorics, graph theory, and homological algebra.

Findings

Confirmed conjecture on torsion primes in homology of poset Lie algebras.

Derived explicit formulas for (co)homology of specific Lie algebra families.

Showed how algebraic Morse theory can be used to construct acyclic matchings and solve combinatorial problems.

Abstract

We investigate the (co)homological properties of two classes of Lie algebras that are constructed from any finite poset: the solvable class $\frak{gl}^\preceq$ and the nilpotent class $\frak{gl}^\prec$. We confirm the conjecture of Jollenbeck that says: every prime power $p^r\!\leq\!n\!-\!2$ appears as torsion in $H_\ast(\frak{nil}_n;\mathbb{Z})$, and every prime power $p^r\!\leq\!n\!-\!1$ appears as torsion in $H_\ast(\frak{sol}_n;\mathbb{Z})$. If $\preceq$ is a bounded poset, then the (co)homology of $\frak{gl}^\preceq$ is \emph{torsion-convex}, i.e. if it contains $p$-torsion, then it also contains $p'$-torsion for every prime $p'\!<\!p$. \par We obtain new explicit formulas for the (co)homology of some families over arbitrary fields. Among them are the solvable non-nilpotent analogs of the Heisenberg Lie algebras from the Cairns & Jambor article, the 2-step Lie algebras from Armstrong & Cairns & Jessup article, strictly block-triangular Lie algebras, etc. The resulting generating functions and the combinatorics of how they are obtained are interesting in their own right. \par All this is done by using AMT (algebraic Morse theory). This article serves as a source of examples of how to construct useful acyclic matchings, each of which in turn induces compelling combinatorial problems and solutions. It also enables graph theory to be used in homological algebra.