[Paper Review] Crossed modules and the integrability of Lie brackets

Iakovos Androulidakis|arXiv (Cornell University)|Jan 7, 2005

Homotopy and Cohomology in Algebraic Topology20 references3 citations

TL;DR

This paper establishes a precise correspondence between the integrability obstruction of transitive Lie algebroids and the lifting obstruction of a naturally associated crossed module of Lie groupoids. It demonstrates that the obstruction to integrating such algebroids is classified by isometablic Čech cohomology, and provides a complete classification of operator extensions of coupling pair crossed modules of PBG-groupoids via a free and transitive action of a cohomology group.

ABSTRACT

We show that the integrability obstruction of a transitive Lie algebroid coincides with the lifting obstruction of a crossed module of groupoids associated naturally with the given algebroid. Then we extend this result to general extensions of integrable transitive Lie algebroids by Lie algebra bundles. Such a lifting obstruction is directly related with the classification of extensions of transitive Lie groupoids. We also give a classification of such extensions which differentiates to the classification of transitive Lie algebroids discussed in \cite{KCHM:new}.

Motivation & Objective

• To identify the integrability obstruction of transitive Lie algebroids with the lifting obstruction of an associated crossed module of Lie groupoids.
• To extend the classification of extensions of integrable transitive Lie algebroids by Lie algebra bundles.
• To provide a cohomological classification of operator extensions for coupling pair crossed modules of PBG-groupoids.
• To relate the integrability problem to geometric prequantization and curvature quantization in symplectic geometry.
• To generalize results from [15] and [17] to the setting of PBG-groupoids and isometablic cohomology.

Proposed method

• Constructs a crossed module of Lie groupoids from a transitive Lie algebroid using its associated Lie groupoid and isotropy bundle.
• Defines isometablic Čech cohomology $\check{H}^1_G(P \times P, ZH)$ for principal bundles with structure group $G$ and fiber $H$.
• Uses local sections and transition functions to define cocycle morphisms $\varphi_{ij}: G \times H \to H$ satisfying $\varphi_{ij}(g)(h_1 h_2) = \varphi_{ik}(g)(h_1) \varphi_{kj}(g)(h_2)$.
• Introduces operator PBG-groupoids as extensions of $P \times P$ by a Lie group bundle $F$, with structure preserved under isometablic transition functions.
• Defines a group action of $\check{H}^1_G(P \times P, ZH)$ on the set of equivalence classes of operator extensions.
• Applies the theory to classify extensions of coupling pair crossed modules of PBG-groupoids via the cohomology group action.

Experimental results

Research questions

• RQ1 What is the precise relationship between the integrability obstruction of a transitive Lie algebroid and the lifting obstruction of its associated crossed module of Lie groupoids?
• RQ2 How can the classification of extensions of integrable transitive Lie algebroids be generalized beyond the case of Lie algebra bundles?
• RQ3 What role does isometablic Čech cohomology play in classifying PBG-groupoids and their extensions?
• RQ4 How does the action of $\check{H}^1_G(P \times P, ZH)$ on operator extensions relate to the geometry of principal bundles and Lie groupoids?
• RQ5 In what way does the obstruction theory for crossed modules of PBG-groupoids generalize the classification of transitive Lie algebroids in [15]?

Key findings

• The integrability obstruction of a transitive Lie algebroid coincides with the lifting obstruction of the associated crossed module of Lie groupoids.
• The set of equivalence classes of operator extensions of a coupling pair crossed module of PBG-groupoids is classified by a free and transitive action of $\check{H}^1_G(P \times P, ZH)$.
• The transition functions $s_{ij}$ of the PBG-groupoid satisfy the isometablic condition $s_{ij}(ug) = \varphi_{ij}(g)(s_{ij}(u))$, where $\varphi_{ij}$ are cocycle morphisms.
• The cohomology group $\check{H}^1_G(P \times P, ZH)$ classifies PBG-groupoids over a principal bundle $P(M,G)$ with fiber type $H$.
• The action of $\check{H}^1_G(P \times P, ZH)$ on operator extensions is well-defined and preserves the structure of the crossed module.
• The theory provides a geometric realization of the prequantization condition for symplectic manifolds as a special case of integrability of transitive Lie algebroids.