Classifying bi-invariant 2-forms on infinite-dimensional Lie groups
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...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.2311.03913 |