Reference

The homotopy Lie algebra of a complex hyperplane arrangement is an algebraic structure associated with the complement of the arrangement. It is defined as the Lie algebra of primitives of a certain Hopf algebra which arises from the cohomology ring of the complement of the hyperplane arrangement. This complement, usually denoted as (X), is the space obtained by removing the union of all hyperplanes in the arrangement from the ambient space (\mathbb{C}^n).

In the specific context of complex hyperplane arrangements, the homotopy Lie algebra encapsulates information about the higher homotopy groups of these complements. It is an essential object in understanding both the topological and algebraic properties of the complement space.

One interesting aspect of these algebras is that they are not necessarily finitely presented, meaning there isn't always a finite set of generators and relations that can describe the algebra completely. For more details on examples where the homotopy Lie algebra is not finitely presented, you can review the research papers available on platforms like Project Euclid and arXiv.

This sophisticated structure is vital for studying and understanding complex hyperplane arrangements' geometry and topology. Here's an illustrative image of the calculation of a homotopy Lie algebra from a hyperplane arrangement setup:

These intricacies highlight the depth and challenges involved in the field of algebraic topology and related mathematical areas.