# An Integration Theorem for Representations of the Tangent Algebroid

Representation Theory of Lie Groupoids

by Geoffrey-Desmond Busche

Date of Examination:2023-11-13

Date of issue:2024-02-09

Advisor:Prof. Dr. Ralf Meyer

Referee:Prof. Dr. Ralf Meyer

Referee:Prof. Dr. Thomas Schick

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## Abstract

### English

This dissertation investigates representations of Lie groupoids and Lie algebroids and the connection between them. Lie groupoids and Lie algebroids are differential-geometric generalisations of Lie groups and Lie algebras. A Lie groupoid representation is a bounded *-homomorphism from the groupoid C∗-algebra of the groupoid into the bounded operators on a separable Hilbert space. A Lie algebroid representation is a unital *-homomorphism from the universal enveloping algebra of the Lie algebroid into the unbounded operators on a separable Hilbert space which has a common, invariant, dense domain. Similar to Lie groups, any Lie groupoid G can be differentiated to a Lie algebroid A(G). In this case, the right-invariant differential operators on G (those differential operators which commute with the multiplication maps rg : h → hg) are a universal enveloping algebra for A(G) and carry a natural involution defined using the divergence for vector fields. I use an algebraic definition of differential operators, which does not involve charts. All of these notions are formally introduced in the first three chapters of this thesis. In Chapter 4 I give a proof of the known fact that every non-degenerate Lie groupoid representation can be differentiated to a representation of its Lie algebroid on the same Hilbert space. I also show that in this derived representation, symmetric differential operators of order 1 act by essentially self-adjoint unbounded operators. Chapter 5 covers measurable fields of Hilbert spaces, which are infinite-dimensional generalisations of vector bundles. I show how every Hilbert space which is the target of a Lie algebroid representation is isomorphic to the section space of a measurable field of Hilbert spaces. Any measurable field of Hilbert spaces H on a space M defines a groupoid of unitary maps U(H) over M. I use this to define a third type of representation, which is a groupoid homomorphism from a Lie groupoid G to the unitary groupoid U(H) of a measurable field of Hilbert spaces over the same base space. I show that each local groupoid homomorphism which is defined on a neighbourhood of the identities can be extended to a global homomorphism if the groupoid has simply connected fibres. Chapters 6 and 7 serve the construction of an integration theorem for Lie algebroid representations. In Chapter 6 I show that exponentials of vector fields act by decomposable unitary operators and use this to integrate representations of the Euclidean tangent bundle TRm to representations of the pair groupoid Rm×Rm. I also demonstrate how to integrate groupoid homomorphisms to representations of the groupoid C∗-algebra. Then I show that the combination of both integration steps is actually inverse to differentiation using explicit computations. Chapter 7 uses techniques introduced in Chapter 6 to prove a generalised integration theorem. This theorem states that every integrable representation of a tangent algebroid TM, where M is a compact, simply connected smooth manifold, can be uniquely integrated to a representation of the pair groupoid M × M. A necessary and sufficient condition for integrability is that all symmetric differential operators of order 1 act by essentially self-adjoint operators and that exponentials of locally commuting vector fields fulfil a local group relation. I show that integration and differentiation are again inverse to each other in this scenario. Finally I investigate a few other conditions for integrability.**Keywords:**Representation; Self-adjoint operator; Differential operator; Lie algebroid; Lie groupoid; *-Algebra