(Co)Homology of Poset Lie Algebras 

We investigate the (co)homological properties of two classes of Lie algebras that are constructed from any finite poset: 
the solvable class gl^\leq and the nilpotent class gl^\le. We confirm the conjecture of Jollenbeck that says: every prime power 
p^r <= n-2 appears as torsion in H_*(nil_n;Z), and every prime power p^r <= n-1 appears as torsion in H_*(sol_n;Z). If \leq is 
a bounded poset, then the (co)homology of gl^\leq is torsion-convex, i.e. if it contains p-torsion, then it also contains 
p'-torsion for every prime p' < p. 

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. 

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.