Namely i propose to consider representation of a group as a sheaf on some geometric object. The groups envisaged here are the bisection group and a group of groupoid selfmaps. We catalogue the primitive ideals of the associated groupoid c. In mathematics, especially in category theory and homotopy theory, a groupoid less often brandt groupoid or virtual group generalises the notion of group in several equivalent ways. Groupoid theory posted by mike shulman this week im at the 20 international category theory conference in syndey, australia, along with our cohosts tom and emily and a. The present paper links the representation theory of lie groupoids and infinitedimensional lie groups. Linking lie groupoid representations and representations. The representation theory of calgebras associated to. When we combine our universal property with the representation theory of commutative c. It has a single eld, which is a principal gbundle on some manifold m.
The representation theory of calgebras associated to groupoids. The theory of characters for finitedimensional representations of finite groupoids is developed and it is shown that irreducible representations of. The result has the structure of a groupoid over the base space x. We shall see that at the level of algebras, this di erence disappears. In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the tannakakrein duality for compact.
Although this representation does not unambiguously descend. On groupoid c algebras, persistent homology and time. For a special class of actions we describe the jacobson topology. Some basic results are presented, and the relation with the. Groupoids in geometric quantization wiskunde radboud. Representations of lie groupoids and infinitedimensional. The picard groupoid in deformation quantization springerlink. In homotopy theory, spaces are glued together fromdiscs. We provide a unified treatment of several results concerning full groups of ample groupoids and paradoxical decompositions attached to them. The 2sphere s2 can be built from a point and a 2disc, but. Representation theory ct, lent 2005 1 what is representation theory. Steinberg, decidability of ow equivalence and isomorphism problems for. Grothendiecks problem homotopy type theory synthetic 1groupoids category theory discs versus morphisms there is a fundamental mismatch. Groups arise in nature as sets of symmetries of an object, which are closed under composition and under taking inverses.
Bg 3 from the fundamental groupoid of mto the groupoid bgwith one object whose automorphism group is g. Crash course on lie groupoid theory school of mathematics. Tannakakrein duality for compact groupoids i, representation theory. In addition to explaining wellknown stuff, well report. Introduction to representation theory mit opencourseware. A double groupoid can also be considered as a generalization of certain higherdimensional groups. It has also led to the omission of topics which are important but not exactly germane to this book. This fall, the socalled quantum gravity seminar at u. Further the stabilizers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid. A classical 1 groupoid is a structured collection ofmorphisms.
On the structure of finite groupoids and their representations mdpi. The value of this point of view is that the tangent. In particular, we relate the representation theory of algebras over rings c ri with an ordered ring r and i2. Tannakakrein duality for compact groupoids i, representation theory massoud amini abstract. And there are two other new things about this seminar. The geometry of squares and their compositions leads to a common representation of a double groupoid in the following diagram. Lie groupoid theory groupoids groupoids a groupoid is a small category where every morphism is an. Groupoids, calgebras and index theory nigel higson lecture at fim, july 10, 2004 1 introduction my goal in this talk is to introduce some topics in alain connes noncommutative geometry, organized around the notion of groupoid and involving for the most part elaborations of the index theory. Besides that, we unify several recent results about paradoxicality in.
Linear representations and frobenius morphisms of groupoids. Just like in ordinary theory of projective representations this means that there is a bona. Pdf imprimitivity theorem for groupoid representations. A vb groupoid is a lie groupoid equipped with a compatible linear structure. We show that smooth representations of lie groupoids give rise to smooth representations of associated lie groups. Let e be a secondcountable, locally compact, hausdorff groupoid equipped with an action of t such that g.
In this paper, we describe a correspondence, up to isomorphism, between vbgroupoids and 2term representations up to homotopy of lie groupoids. Regauging groupoid, symmetries and degeneracies for. Introduction to representation theory mit mathematics. Group with a partial function replacing the binary. A systematic development of the fundamentals of the theory of groupoid calgebras was provided by jean renault in 17, which is the classical reference for the. The representation theory of nite groups has a long history, going back to the 19th century and earlier. Then, as chris points out, the abstract representation theory of groupoids essentially reduces to the representation theory of groups. The notion of group action can be put in a broader context by using the action groupoid. Vbgroupoids and representation theory of lie groupoids. Analogous to the fundamental groupoid it is possible to get rid of the choice of a base point and to define a monodromy groupoid. Since gis nite, a principal gbundle on mis precisely a gcover, which is in turn precisely a groupoid homomorphism 1m. A linear representation is a representation on a category of vector spaces or similar vect, mod, etc. The representation theory of groups is an established branch of mathematics. We interpret such a representation as a module over the twisted drinfeld double d g.
Here we consider homotopy classes of lifts of paths in the base space x of a fibration. First, it will be jointly run by john baez and james dolan. The quintessential example might be the symmetry group of a square. One sometimes considers representations on objects other than linear spaces such as permutation representations but often these are called not representations but actions. Prior to this there was some use of the ideas which we can now identify as representation theory characters of cyclic groups as used by. For example, the symmetric group s n is the group of all. In a series of papers, we have shown that from the representationtheory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the tannakakrein duality for compact groups. Mobius functions and semigroup representation theory 5 let abe a unital ring and ga groupoid.
Thus every smooth representation of a lie groupoid gives rise to a joint smooth representation of in. I have tried to include enough examples along the way, together. We consider the deaconurenault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact hausdorff space. I have also skimmed over some of the more technical aspects of groupoid theory for example the question of amenability. The evensluweinstein approach uses the fact that, if g mis a lie groupoid with lie algebroid a, then there is a natural representation of the 1jet prolongation groupoid j1gon the 2term complex a. Representations of groupoids the representation theory of groups is an established branch of mathematics. Then, representations of the lie groupoids give rise to representations of the. Surfaces and the representation theory of nite groups. In this theory, one considers representations of the group algebra a cg of a. Pdf in this paper, both the structure and the theory of representations of finite groupoids are discussed. Continuous representations of groupoids mathematics. As another example consider the representation theory of quivers. Our motivation is to introduce the main ideas that underpin techniques from the theory of lin ear representation of groupoids in relation with. Representation theory is the study of the concrete ways in which abstract groups can be realized as groups of rigid transformations of r nor c.
Pdf on the structure of finite groupoids and their representations. A finitedimensional unitary representation of a group admits an orthogonal decomposition into irreducible unitary subrepresentations. A symmetry of the square is any rigid motion of euclidean space which preserves the square. Hilbert spaces, carrying a representation of the groupoid. Pdf the representation theory of calgebras associated. Under this correspondence, the tangent bundle of a lie groupoid g corresponds to the adjoint representation of g. Regular representation of groupoid c algebras and applications to. This includes a criterion for the full group of an ample groupoid being amenable as well as comparison of its orbit, koopman and groupoid leftregular representations.
Riverside will actually tackle geometric representation theory the marvelous borderland where geometry, groupoid theory and logic merge into a single subject. Since their introduction in 1980, groupoid \c\algebras have been intensively studied with diverse applications, including graph algebras, classification theory, variations on the baumconnes conjecture, and noncommutative geometry. Et is a principal groupoid with haar system \lambda. In part ii we study such representations of groupoids on continuous. Translation groupoids and orbifold cohomology fort lewis college. Representation theory university of california, berkeley. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. The representation theory of calgebras associated to groupoids article pdf available in mathematical proceedings of the cambridge philosophical society 15301 july 2010 with 31 reads.