# Holonomy Lie algebra of a geometric lattice

## TL;DR

This paper introduces a combinatorial definition of the holonomy Lie algebra for finite geometric lattices, extending known results from hyperplane arrangements to a broader class of lattices.

## Contribution

It defines the holonomy Lie algebra for geometric lattices and describes its structure for hypersolvable and supersolvable lattices, generalizing previous work on hyperplane arrangements.

## Key findings

- Holonomy Lie algebra of a solvable pair decomposes as an almost-direct product.
- Structure of holonomy Lie algebra for hypersolvable lattices is characterized.
- Applications to supersolvable oriented matroids and arrangements are provided.

## Abstract

Motivated by Kohno's result on the holonomy Lie algebra of a hyperplane arrangement, we define the holonomy Lie algebra of a finite geometric lattice in a combinatorial way. For a solvable pair of lattices, we show that the holonomy Lie algebra is an almost-direct product of the holonomy Lie algebra of the sublattice and a free Lie subalgebra. This yields the structure of the holonomy Lie algebra of a finite hypersolvable (including supersolvable) lattice. As applications, we obtain the structure of the holonomy Lie algebra of (the Salvetti complex of) a supersolvable oriented matroid, and that of a hypersolvable arrangement.