[Paper Review] Abelian extensions of infinite-dimensional Lie groups

Karl‐Hermann Neeb|ArXiv.org|Feb 18, 2004

Advanced Topology and Set Theory46 references43 citations

TL;DR

This paper develops a cohomological framework for abelian extensions of infinite-dimensional Lie groups modeled on locally convex spaces, introducing the cohomology group $ H^2_s(G,A) $ for locally smooth cochains. It establishes an exact sequence linking group cohomology to Lie algebra cohomology and characterizes integrability obstructions via period and flux homomorphisms, with applications to diffeomorphism groups and geometric structures like principal bundles.

ABSTRACT

In the present paper we study abelian extensions of connected Lie groups $G$ modeled on locally convex spaces by smooth $G$-modules $A$. We parametrize the extension classes by a suitable cohomology group $H^2_s(G,A)$ defined by locally smooth cochains and construct an exact sequence that describes the difference between $H^2_s(G,A)$ and the corresponding continuous Lie algebra cohomology space $H^2_c(\g,\a)$. The obstructions for the integrability of a Lie algebra extensions to a Lie group extension are described in terms of period and flux homomorphisms. We also characterize the extensions with global smooth sections resp. those given by global smooth cocycles. Finally we apply the general theory to extensions of several types of diffeomorphism groups.

Motivation & Objective

• To develop a cohomological classification of abelian extensions of connected Lie groups modeled on locally convex spaces.
• To define and study the cohomology group $ H^2_s(G,A) $ using locally smooth cochains for smooth $ G $-modules $ A $.
• To describe the obstruction to integrating Lie algebra extensions to global Lie group extensions via period and flux homomorphisms.
• To characterize extensions with global smooth sections or arising from global smooth cocycles.
• To apply the theory to diffeomorphism groups and geometric structures such as principal bundles and prequantization.

Proposed method

• Introduce the cohomology group $ H^2_s(G,A) $ using locally smooth cochains to classify abelian extensions of Lie groups.
• Construct an exact sequence relating $ H^2_s(G,A) $ to the continuous Lie algebra cohomology $ H^2_c(rak{g},rak{a}) $.
• Define period and flux homomorphisms to characterize the integrability of Lie algebra extensions to Lie group extensions.
• Use the cup product structure on group cochains $ C^p_s(G,U) \times C^q_s(G,V) \to C^{p+q}_s(G,W) $ to define a product on cohomology.
• Establish compatibility between group cohomology and Lie algebra cohomology via the differentiation map $ D $, showing $ D(\alpha \cup \beta) = D\alpha \wedge D\beta $.
• Apply the framework to specific examples, including diffeomorphism groups of manifolds and modules of $ \lambda $-densities.

Experimental results

Research questions

• RQ1 How can abelian extensions of infinite-dimensional Lie groups be classified using cohomology?
• RQ2 What is the relationship between the smooth group cohomology $ H^2_s(G,A) $ and the continuous Lie algebra cohomology $ H^2_c(\frak{g},\frak{a}) $?
• RQ3 What obstructions prevent a Lie algebra extension from integrating to a Lie group extension?
• RQ4 When do abelian extensions admit global smooth sections or arise from global smooth cocycles?
• RQ5 How can this theory be applied to diffeomorphism groups and geometric structures like principal bundles?

Key findings

• The cohomology group $ H^2_s(G,A) $ parametrizes abelian extensions of Lie groups $ G $ by smooth $ G $-modules $ A $, using locally smooth cochains.
• An exact sequence is constructed that relates $ H^2_s(G,A) $ to the continuous Lie algebra cohomology $ H^2_c(\frak{g},\frak{a}) $, clarifying the difference between group and algebraic extensions.
• The integrability of a Lie algebra extension to a Lie group extension is obstructed by the period and flux homomorphisms, which vanish if and only if the extension integrates.
• Extensions with global smooth sections correspond exactly to those arising from global smooth cocycles, establishing a cohomological characterization.
• The cup product on group cochains induces a well-defined product on cohomology, compatible with the Lie algebra cup product under differentiation.
• The theory is applied to diffeomorphism groups, including the group of volume-preserving diffeomorphisms and the circle diffeomorphism group with $ \lambda $-density modules, yielding geometrically meaningful extensions.