Automorphic Lie Algebras and Cohomology of Root Systems

TL;DR

This paper develops a cohomology theory for root systems that aids in understanding Automorphic Lie Algebras, providing models, properties, and applications to Lie algebra contractions and filtrations.

Contribution

It defines a new cohomology framework for root systems, explores its properties, and demonstrates its relevance to Automorphic Lie Algebras and Lie algebra contractions.

Findings

Defined cohomology for root systems

Established properties like cup product and symmetry restrictions

Connected cohomology to Lie algebra contractions and filtrations

Abstract

A cohomology theory of root systems emerges naturally in the context of Automorphic Lie Algebras, where it helps formulating some structure theory questions. In particular, one can find concrete models for an Automorphic Lie Algebra by integrating cocycles. In this paper we define this cohomology and show its connection with the theory of Automorphic Lie Algebras. Furthermore, we discuss its properties: we define the cup product, we show that it can be restricted to symmetric forms, that it is equivariant with respect to the automorphism group of the root system, and finally we show acyclicity at dimension two of the symmetric part, which is exactly what is needed to find concrete models for Automorphic Lie Algebras. Furthermore, we show how the cohomology of root systems finds application beyond the theory of Automorphic Lie Algebras by applying it to the theory of contractions and filtrations of Lie algebras. In particular, we show that contractions associated to Cartan $\mathbb{Z}$-filtrations of simple Lie algebras are classified by $2$-cocycles, due again to the vanishing of the symmetric part of the second cohomology group.