Geometric representation theory is a very active field, at the center of recent advances in number theory and theoretical physics. 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. We recall the classical action of the heisenberg algebra h on the usual and equivariant homology of the hilbert scheme of points on c2, due to grojnowskinakajima. It is interesting to note that while representation theory of quantum groups or a. As already noted, in our framework, the existence of a good amodel for yis supposed to ensure that deformation quantization of yproduces an actual deformation of the algebra of holomorphic functions, depending on a complex parameter or 1n, not just a formal.
This classic monograph provides an overview of modern advances in representation theory from a geometric standpoint. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and its algebraic operations for example, matrix. These are the class s theories, and some of them have speci c names. Here you can learn about derived categories and perverse sheavesdmodules which are essential tools to study geometric representation theory and how they are connected to representation theory. Representation theory, geometric langlands duality and. A geometricallyoriented treatment of the subject is very timely and has long been desired, especially since the discovery of dmodules in the early 1980s and the quiver approach to quantum groups in the early 1990s. The eld grew from a sequence of signi cant discoveries of connections among dmodules, constructible sheaves, and representation theory, including the microlocal studies. The spinor representation as we have seen, the groups spinn have a representation on rn given by identifying v. Geometric construction of the enveloping algebra uslnc 193 4. One of the reasons for interest in geometric representation theory, of which the above is our first example, is that it produces representations with canonical bases. The stud y of geometric prop ertie s of state spaces is a basic topic in the theory of op erator algebras see, e. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Geometric representation theory connections for women, august 2829 introductory workshop, september 25 msri, 2014 1 t.
When proofs are not included, references for them are given. Nineteenthcentury representation theory helped to explain the structure of electron orbitals, and 1920s representation theory is at the heart of quantum chromodynamics. Such central extension gives rise to a goup scheme that leads to a geometric construction of unrestricted representations. Let us see what verma modules look like explicitly in the simplest case of g sl 2. Say someone is familiar with algebraic geometry enough to care about things like gbundles, and wants to talk about vector bundles with structure group g, and so needs to know representation theory, but wants to do it as geometrically as possible. Arapura, notes on dmodules and connections with hodge theory geometric representation theory geometric langlands seminar webpage v. The principal goal of the introductory workshop will be to provide a gateway for graduate students and new postdocs to the rich and exciting, but potentially daunting, world of geometric representation theory. Geometric quantization and representation theory 5 lemma 1. Theory x will be conformal in the 2manifold but topological in the 4manifold. Representation theory is the study of the basic symmetries of mathem. The process of fitting the model to the target structure is guided by physical principles and. So representation theory has kind of an intimidating feel to it for an outsider.
Shrawan kumar, borelweilbott theorem and geometry of schubert varieties. Request pdf enveloping algebras and geometric representation theory since its inception in the early seventies, the study of enveloping algebras has undergone a significant and continuous. Geometric representation theory of restricted lie algebras of classical type. Geometric representation theory lies in the crossroads of a number of di erent elds, including representation theory, algebraic geometry, number theory and symplectic geometry. Geometric representation theory studies representations of various symmetry objects like algebraic groups, hecke algebras, quantum groups, quivers etc. This volume provides a selfcontained overview of some of the recent advances in representation theory from a geometric standpoint. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear. Symplectic approaches in geometric representation theory. Bernhard preim, charl botha, in visual computing for medicine second edition, 2014. Representation theory is the study of the basic symmetries of mathematics and physics. It has good description of geometric representation theory. Representation theory and geometry university of sydney. Geometric representation theory lecture 19 we might still perform a good theory for small categories by considering todds remarks about the beckchevalley property.
The seminar is jointly run by john baez and james dolan. Lectures on geometric group theory cornelia drutu and michael kapovich. In mathematics geometric invariant theory or git is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. Geometric representation theory of restricted lie algebras. Representation theory studies how algebraic structures act on objects. It was developed by david mumford in 1965, using ideas from the paper hilbert 1893 in classical invariant theory geometric invariant theory studies an action of a group g on an algebraic variety or scheme x and provides. Pdf geometric representation theory for unitary groups.
E is planar, if it can be drawn in the plane so that its edges are jordan curves and they intersect only at their endnodes1. Kirill zainoulline, from the university of ottawa titles and abstracts of his talks can be found here. The representation theory of nite groups has a long history, going back to the 19th century and earlier. An early profound achievement was the famous conjecture by kazhdanlusztig about characters of highest weight modules over a complex semisimple lie algebra, and its subsequent proof by beilinsonbernstein and brylinskikashiwara.
This will be an ongoing learning seminar devoted to current research directions where infinitesimal symmetries play a distinguished role. Geometric representation theory lecture 19 the ncategory. We quantize the maxwell chern simons theory in a geometric representation that generalizes the abelian loop representation. I had two books in hand, firstly representation theory of finite groups, an introductory approach by benjamin steinberg, and secondly serres linear representations of finite groups.
Wilfried schmid notes by matvei libine, geometric methods in representation theory. Geometric representation theory lecture 1 the ncategory cafe. Representation theory, geometric langlands duality and categori cation joel kamnitzer october 21, 2014 abstract the representation theory of reductive groups, such as the group gl n of invertible complex matrices, is an important topic, with applications to number theory, algebraic geometry, mathematical physics, and quantum topology. Preface the goal of this book is to present several central topics in geometric group theory,primarilyrelatedtothelargescalegeometryofin. For the standard measure on the sphere of radius r, bk2. Geometric representation theory studies representations of various symmetry objects like algebraic. The aim of the summer school is to provide minicourses on active themes in geometric representation theory, including those mentioned above. Representation theory and complex geometry neil chriss. Additionally to peter crooks answer i would recommend to study the book of hotta and others. Introductory references for geometric representation theory.
In addition, there will be research talks on recent progress in the field, and a poster session featuring work of graduate students. Jun 27, 2016 i found an article related to geometric representation theory. Borel subgroups, parabolic subgroups, the bruhat decomposition 10 4. They comprise three contributions, due to alexander braverman and michael finkelberg, andrei negut, and alexei oblomkov, respectively. Enumerative geometry and geometric representation theory. Deformable models are based on a flexible geometric representation, such as bsplines, which provide the necessary degrees of freedom to adapt the model to a large variety of shapes. Geometric representation theory seminar fall 2007 john baez and james dolan this fall, our seminar is tackling geometric representation theory the marvelous borderland where geometry, groupoid theory and logic merge into a single subject. Geometric representation an overview sciencedirect topics. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions.
Geometric representation theory of the hilbert schemes part i alexander tsymbaliuk abstract. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. Frenkel, langlands correspondence for loop groups, cambridge studies in advanced mathematics 103, cambridge university press, 2007, isbn 978. Introduction to representation theory by pavel etingof. Later on, we shall study some examples of topological compact groups, such as u1 and su2. I found an article related to geometric representation theory. Written for the 2015 algebraic geometry summer institute. Geometric representation theory and gauge theory cetraro. Geometric representation theory is a young but fast developing research area at the intersection of these subjects.
This book is an introduction to geometric representation theory. Geometric representation theory jon cohen lecture 1. The notes are based on lectures delivered at the cime school geometric representation theory and gauge theory held in cetraro, italy, in june 2018. I studied representation theory for the first time 3 months ago. For a classical semisimple lie algebra, we construct equivariant line bundles whose global sections afford representations with a nilpotentpcharacter. The gns construction produ c es repr e sen tatio ns of op erato r. Equivariant cohomology and divided differences 5 3. Geometry of moduli spaces and representation theory. Representation theory was born in 1896 in the work of the ger man mathematician. Enveloping algebras and geometric representation theory. Representation theory and complex geometry modern birkhauser. When t2 we get the topological twist of n 4 super yangmills. A learning roadmap for representation theory mathoverflow.
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