An Integration Theorem for Representations of the Tangent Algebroid ; Representation Theory of Lie Groupoids

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 ...