Higher representation theory in algebra and geometry. Geometric methods in representation theory of hecke. These lectures are mainly based on, and form a condensed survey of the book by n. First,these algebras are important for statisyical mechanics, conformal. This monograph provides an introduction to the theory of clifford algebras, with an emphasis on its connections with the theory of lie groups and lie algebras. Various algebras arising naturally in representation theory such as the group algebra of a weyl group, the universal enveloping algebra of a complex semisimple lie algebra, a quantum group or the iwahorihecke algebra of. Representations of reduced enveloping algebras and cells in the. Various algebras arising naturally in representation theory such as the group algebra of a weyl group, the universal enveloping algebra of a complex semisimple lie algebra, a quantum group or the iwahorihecke algebra of biinvariant functions under convolution on a padic group, are considered. The noncommutative algebras of interest include algebras of differential operators, enveloping algebras, and quantum groups. Representation theory of groups, quantum groups, and. Universal enveloping algebras of lie algebras appeared more than a century ago as one of the major tools in lie theory. Among the results that i cover from algebraic geometry. A as well as the entire analytic elementsc a carry natural topologies making them algebras with ac enveloping algebra. In mathematics, a universal enveloping algebra is the most general unital, associative algebra that contains all representations of a lie algebra.
Geometric interpretation of universal enveloping algebras. Algebras and representation theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including lie algebras and superalgebras, rings of differential operators, group rings and algebras, c algebras and hopf algebras, with particular emphasis on quantum groups. If we knew differential geometry, we would say that d. Schemy issues rarely enter geometric representation theory. Enveloping algebras and geometric representation theory enveloping algebras and geometric representation theory. Representation theory of lie superalgebras and related topics. Very roughly speaking, representation theory studies symmetry in linear spaces. In addition, a number of things about algebric groups, lie algebras and their represen. We propose a generalisation to quasitilted and almost hereditary algebras, introducing two. The existence of this universal object goes back to rosenlicht 43, but see also the more modern and general algebrogeometric treatment in 35. Suppose that g is the lie algebra of a lie group g.
Enveloping algebras and geometric representation theory 7 example. Hopf algebra structure on the universal enveloping algebra of a leibniz algebra. We study universal enveloping hopf algebras of lie algebras in the category of weakly complete vector spaces over the real and complex field. The goal of this dissertation is to exploit connections between algebraic groups and quantized enveloping algebras in order to study the cohomology of certain nitedimensional hopfsubalgebras of u g, called the frobeniuslusztig.
This support is the set of integers isuch that hif 6 0, where hiis the cohomological functor associated with the usual tstructure. A nice short text is the book \lectures on lie groups and lie algebras by. Introduction to representation theory mit opencourseware. This lie algebra is a quite fundamental object, that crops up at many places, and thus its representations are interesting in themselves. The aims of this course are to introduce techniques for understanding finite and infinite dimensional representations, and to understand the structure of the enveloping algebra using representation theory. Universal enveloping algebras are used in the representation theory of lie groups and lie algebras. Clifford algebras and lie theory eckhard meinrenken. Adopting a panoramic viewpoint, this book offers an introduction to four different flavors of representation theory. A representation of an associative algebra aalso called a left amodule is a vector space v equipped with a homomorphism a. Crossed products, the mackeyrieffelgreen machine and applications lecture 1 induced representations of groups in the sense of mackey and blattner. Geometric methods in the representation theory of hecke.
On weakly complete universal enveloping algebras of pro. Homology of associative algebras ordinary and restricted cohomology of restricted lie algebras. Cohomology of frobeniuslusztig kernels of quantized. Quite generally, we study this class of algebras from the point of view of poisson geometry, exhibiting connections between their representation theory and some wellknown. Introduction to representation theory pavel etingof, oleg golberg, sebastian hensel, tiankai liu, alex schwendner, dmitry vaintrob, and elena yudovina.
Assock to liek, and it has a left adjoint u called universal enveloping algebra. Topics of particular interest include noncommutative projective algebraic geometry, noncommutative resolutions of commutative or noncommutative singularities,calabiyau algebras, deformation theory and poisson structures, as well as the interplay of these subjects with the algebras appearing in representation theory like enveloping algebras. Introduction to representation theory by pavel etingof. Quiver varieties, and more generally symplectic resolutions, precipitate geometric realizations of various noncommutative algebras and lead to a deeper understanding of the representation theory of these algebras. The structures involved can be generalized to quantum groups and kacmoody lie algebras. These are the notes of a topics in representation theory class i taught in princeton. In representation theory they appear as the images of the associated representations of the lie algebras or of the enveloping algebras on the garding domain and in quantum field theory they occur as the vector space of field operators or the algebra generated by them.
Okounkov, is a geometric approach to representation theory of elliptic dynamical quantum groups associated with quiver varieties. Representation theory of groups, quantum groups, and operator algebras university of copenhagen 15 june 2015 minicourses siegfried echterhoff title. Convolution algebras provide a uniform approach to the construction of many familiar objects such as group algebras of weyl groups, a ne hecke algebras, degenerate a ne hecke algebras as well as quotients of universal enveloping algebras and quantized loop algebras. Two other recommendable texts which only discuss lie algebras are the books \introduction to lie algebras and representation theory by j. Actually e and f induce an action of the quantized universal enveloping algebra u qsl 2, a. Enveloping algebras and geometric representation theory. Geometric methods in representation theory of hecke algebras. Representation theory of finite dimensional lie algebras. So the geometry of the universal enveloping algebra is really the geometry of coadjoint orbits.
In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by lusztig led to the first general construction of pbw bases of these algebras. This idea was successfully developed by dixmier 7 and moeglin 35 in the case of enveloping algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and clifford algebras. Introduction to lie algebras and representation theory. Ginzburg, representation theory and complex geometry, birkhauser 1997. There is more than one version of what should be the universal enveloping algebra of a leibniz algebra. Participating in or organising an event types participation in or organisation af a conference. Indeed, in the semisimple case, the adjoint representation. The elliptic stable envelopes, defined recently by m. Representation theory of geometric extension algebras 5 proof. A representation of an associative algebra aalso called a left amodule is a vector space v equipped with a homomorphism a endv, i. Summer school on geometric representation theory by asilata.
This description points to two obvious generalizations. We proceed by induction on the width of the support of f. Ideals in the enveloping algebra of the positive witt algebra. Geometric construction of the enveloping algebra uslnc 193 4. Enveloping algebras and geometric representation theory mfo. Various algebras arising naturally in representation theory such as the group algebra of a weyl group, the universal enveloping algebra of a complex semisimple lie algebra, a quantum group or the iwahorihecke algebra. Various algebras arising naturally in representation theory such as the group algebra of a weyl group, the universal enveloping algebra of a. Representation theory and complex geometry 1997 birkhauser boston basel berlin. Representation theory of lie superalgebras and related topics generalized teichmuller spaces, spin structures, and ptolemy transformations ivan chiho, ip kyoto university abstract. Mathematisches forschungsinstitut oberwolfach enveloping. Teichmuller space is a fundamental space that is important in many areas of mathematics and physics. Universal envelopping algebras, levis theorem, serres theorem, kacmoody lie algebra, the kostants form of the envelopping algebra and a beginning of a proof of the chevalleys theorem.
Hopf algebra structure on the universal enveloping algebra. Indeed, most of the books listed in the references section devote a substantial fraction of their pages to representation theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. Various algebras arising naturally in representation theory such as the group algebra of a weyl group, the universal enveloping algebra of a complex. From the work of ringel r it is known that the algebras u, ucan be reconstructed purely in terms of the representation theory of this quiver, in the case of graphs of type. A separate part of the book is devoted to each of these areas and they are all treated in sufficient depth to enable and hopefully entice the reader to pursue. Although ados theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given lie algebra. Modular lie algebras universal enveloping algebras representations of associative algebras auslanderreiten theory module varieties support varieties, rank varieties, ppoints homological algebra. In this paper, the geometric method of l2 is extended to the case of. Enumerative geometry and geometric representation theory. Unbounded operator algebras and representation theory.
A subrepresentation of a representation vis a subspace u. The second chapter brings the structure of the semisimple lie algebras. With spela spenko, michaela vancliff, padmini veerapen, and. Representation theory of semisimple lie algebras 2009. They find many applications in differential geometry and mathematical physics, and are indispensable to many directions of research in representation theory of lie groups and algebras. Geometric methods in the representation theory of hecke algebras. Kraft, geometric methods in representation theory, in sln 944. Associating geometry to the lie superalgebra sl11 and to the color lie algebra sl c 2,k. The enveloping algebras seem to be very interesting objects for several reasons.
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