Let e be a secondcountable, locally compact, hausdorff groupoid equipped with an action of t such that g. Pdf on the structure of finite groupoids and their representations. Thus every smooth representation of a lie groupoid gives rise to a joint smooth representation of in. The representation theory of calgebras associated to groupoids. In homotopy theory, spaces are glued together fromdiscs. The picard groupoid in deformation quantization springerlink. 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.
Introduction to representation theory mit opencourseware. Riverside will actually tackle geometric representation theory the marvelous borderland where geometry, groupoid theory and logic merge into a single subject. I have also skimmed over some of the more technical aspects of groupoid theory for example the question of amenability. Bg 3 from the fundamental groupoid of mto the groupoid bgwith one object whose automorphism group is g. A classical 1 groupoid is a structured collection ofmorphisms. It has also led to the omission of topics which are important but not exactly germane to this book. We shall see that at the level of algebras, this di erence disappears. Namely i propose to consider representation of a group as a sheaf on some geometric object. Introduction to representation theory mit mathematics. In particular, we relate the representation theory of algebras over rings c ri with an ordered ring r and i2. Regauging groupoid, symmetries and degeneracies for. 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. For a special class of actions we describe the jacobson topology. Besides that, we unify several recent results about paradoxicality in.
The geometry of squares and their compositions leads to a common representation of a double groupoid in the following diagram. The theory of characters for finitedimensional representations of finite groupoids is developed and it is shown that irreducible representations of. In part ii we study such representations of groupoids on continuous. Some basic results are presented, and the relation with the. Representation theory ct, lent 2005 1 what is representation theory. Although this representation does not unambiguously descend. In addition to explaining wellknown stuff, well report. Linear representations and frobenius morphisms of 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. Pdf imprimitivity theorem for groupoid representations. A finitedimensional unitary representation of a group admits an orthogonal decomposition into irreducible unitary subrepresentations.
A double groupoid can also be considered as a generalization of certain higherdimensional groups. The value of this point of view is that the tangent. Crash course on lie groupoid theory school of mathematics. Here we consider homotopy classes of lifts of paths in the base space x of a fibration. The representation theory of calgebras associated to. On the structure of finite groupoids and their representations mdpi. Group with a partial function replacing the binary. Linking lie groupoid representations and representations. 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. This fall, the socalled quantum gravity seminar at u.
Steinberg, decidability of ow equivalence and isomorphism problems for. I have tried to include enough examples along the way, together. First, it will be jointly run by john baez and james dolan. A vb groupoid is a lie groupoid equipped with a compatible linear structure. The 2sphere s2 can be built from a point and a 2disc, but. Lie groupoid theory groupoids groupoids a groupoid is a small category where every morphism is an. Hilbert spaces, carrying a representation of the groupoid. Tannakakrein duality for compact groupoids i, representation theory massoud amini abstract. The notion of group action can be put in a broader context by using the action groupoid. The representation theory of groups is an established branch of mathematics. As another example consider the representation theory of quivers. Regular representation of groupoid c algebras and applications to.
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. Translation groupoids and orbifold cohomology fort lewis college. Representation theory university of california, berkeley. The groups envisaged here are the bisection group and a group of groupoid selfmaps. Grothendiecks problem homotopy type theory synthetic 1groupoids category theory discs versus morphisms there is a fundamental mismatch. In this paper, we describe a correspondence, up to isomorphism, between vbgroupoids and 2term representations up to homotopy of lie groupoids. On groupoid c algebras, persistent homology and time. One sometimes considers representations on objects other than linear spaces such as permutation representations but often these are called not representations but actions. We consider the deaconurenault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact hausdorff space. Tannakakrein duality for compact groupoids i, representation theory. Vbgroupoids and representation theory of lie groupoids. Then, representations of the lie groupoids give rise to representations of the. The quintessential example might be the symmetry group of a square. We provide a unified treatment of several results concerning full groups of ample groupoids and paradoxical decompositions attached to them.
The representation theory of nite groups has a long history, going back to the 19th century and earlier. Pdf the representation theory of calgebras associated. In this theory, one considers representations of the group algebra a cg of a. Just like in ordinary theory of projective representations this means that there is a bona. Since gis nite, a principal gbundle on mis precisely a gcover, which is in turn precisely a groupoid homomorphism 1m. Groups arise in nature as sets of symmetries of an object, which are closed under composition and under taking inverses. A systematic development of the fundamentals of the theory of groupoid c algebras was provided by jean renault in 17, which is the classical reference for the. 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. Mobius functions and semigroup representation theory 5 let abe a unital ring and ga groupoid. Pdf in this paper, both the structure and the theory of representations of finite groupoids are discussed. A linear representation is a representation on a category of vector spaces or similar vect, mod, etc. 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. We catalogue the primitive ideals of the associated groupoid c.
Groupoids in geometric quantization wiskunde radboud. 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. 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. Under this correspondence, the tangent bundle of a lie groupoid g corresponds to the adjoint representation of g. Surfaces and the representation theory of nite groups. Then, as chris points out, the abstract representation theory of groupoids essentially reduces to the representation theory of groups. Et is a principal groupoid with haar system \lambda.
A symmetry of the square is any rigid motion of euclidean space which preserves the square. For example, the symmetric group s n is the group of all. Representations of groupoids the representation theory of groups is an established branch of mathematics. 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. It has a single eld, which is a principal gbundle on some manifold m. We interpret such a representation as a module over the twisted drinfeld double d g. 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. Further the stabilizers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid.
The present paper links the representation theory of lie groupoids and infinitedimensional lie groups. When we combine our universal property with the representation theory of commutative c. We show that smooth representations of lie groupoids give rise to smooth representations of associated lie groups. 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. The result has the structure of a groupoid over the base space x. 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.
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