Homology, Homotopy and Applications

Volume 19 (2017)

Number 2

(Co)homology of poset Lie algebras

Pages: 1 – 20

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n2.a1

Authors

Leon Lampret (Institute for Mathematics, Physics and Mechanics, Ljubljana, Slovenia; and Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Slovenia)

Aleš Vavpetič (Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Slovenia)

Abstract

We investigate (co)homological properties of Lie algebras that are constructed from a finite poset: the solvable class $\mathfrak{gl}^\preceq$ and the nilpotent class $\mathfrak{gl}^\prec$. We confirm the conjecture [reference 8, 1.16(1), p. 141] that says: every prime power $p_r \leqslant n-2$ appears as torsion in $H_{*} (\mathfrak{nil}_n ; \mathbb{Z})$, and every prime power $p_r \leqslant n-1$ appears as torsion in $H_{*} (\mathfrak{sol}_n ; \mathbb{Z})$. If $\preceq$ is a bounded poset, then the (co)homology of $\mathfrak{gl}^\preceq$ is torsion-convex, i.e., if it contains $p$-torsion, then it also contains $p^{\prime}$-torsion for every prime $p^{\prime} \lt p$.

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 [reference 2], the 2-step Lie algebras from [reference 1], strictly block-triangular Lie algebras, etc. The combinatorics of how the resulting generating functions are obtained are interesting in their own right.

All this is done by using AMT (algebraic Morse theory [references 9, 12, 8]). 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.

Keywords

algebraic/discrete Morse theory, homological algebra, chain complex, acyclic matching, solvable Lie algebra, triangular matrix, torsion table, algebraic combinatorics

2010 Mathematics Subject Classification

13D02, 13P20, 17B56, 18G35, 55-04, 58E05

Received 30 April 2015

Published 19 July 2017