Classifying bi-invariant 2-forms on infinite-dimensional Lie groups

TL;DR

This paper generalizes the classification of bi-invariant 2-forms from compact Lie groups to all finite-dimensional and certain infinite-dimensional Lie groups, revealing new structural insights and correcting previous assumptions.

Contribution

It extends the classification of bi-invariant 2-forms to arbitrary finite-dimensional and Milnor regular infinite-dimensional Lie groups, beyond the known cohomological framework.

Findings

Bi-invariant 2-forms are determined by Ad-invariant bilinear forms.

For certain diffeomorphism groups, the space of bi-invariant 2-forms is finite-dimensional.

The classification invalidates previous assumptions about forms on PU(H).

Abstract

A bi-invariant differential 2-form on a Lie group G is a highly constrained object, being determined by purely linear data: an Ad-invariant alternating bilinear form on the Lie algebra of G. On a compact connected Lie group these have an known classification, in terms of de Rham cohomology, which is here generalised to arbitrary finite-dimensional Lie groups, at the cost of losing the connection to cohomology. This expanded classification extends further to all Milnor regular infinite-dimensional Lie groups. I give some examples of (structured) diffeomorphism groups to which the result on bi-invariant forms applies. For symplectomorphism and volume-preserving diffeomorphism groups the spaces of bi-invariant 2-forms are finite-dimensional, and related to the de Rham cohomology of the original compact manifold. In the particular case of the infinite-dimensional projective unitary group PU(H) the classification invalidates an assumption made by Mathai and the author about a certain 2-form on this Banach Lie group.