Status of the theory of infinitedimensional representations of quantum algebras in the case of infinite dimensional representations of real semisimple lie algebras, there is a onetoone correspondence between irreducible infinite dimensional representations of a connected simply connected real semisimple lie group g and those of its lie. Ottesen that we provide will stimulate you to be smarter. Ergebnissedermathematik volume51 undihrergrenzgebiete. Continuing on, the main subjects of the book are the structure real and imaginary root systems of and the. In order to apply them in theoretical and mathematical physics one has to develop the theory of representations of these groups and algebras. Pdf ebook infinite dimensional groups and algebras in. Click download or read online button to get infinite dimensional lie algebras an introduction progress in mathematics book now. Hilbert schemes of points and infinite dimensional lie. The representation theory of infinite dimensional groups is an important tool for studying conformal field theory, problems in statistical mechanics, and string theory. Author links open overlay panel roberto floreanini a b roberto percacci c a ergin sezgin d. Quantum cluster algebras and quantum nilpotent algebras pnas. Finite dimensional algebras and quantum groups american.
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. Finite dimensional algebras, quantum groups and finite groups of lie type jie du abstract. Theweylkaccharacter formula and jacobiriemann theta functions 115. Infinite dimensional lie algebras an introduction progress. The tensor algebra is infinitedimensional, even if v is finitedimensional. In more recent times, these interrelations have become even more strikingly apparent. Quantum and noncommutative analysis past, present and.
Lectures in lie groups and quantum groups theo johnsonfreyd. Poissonlie structures on infinite dimensional jet groups and quantum groups related to them j. Infinite dimensional algebras and their applications to quantum integrable systems. Infinitedimensional lie algebras in 4d conformal quantum. We shall discuss generic extension monoids associated with. Applications include fluid dynamics, maxwells equations, and plasma physics. Isomorphism classes of small quantum groups in positive. Infinite dimensional algebras and their applications to quantum integrable systems article pdf available in journal of physics a mathematical and theoretical 4119. Representations of w algebras and frenkelkacwakimoto conjecture.
In addition to the traditional instructional workshop preceding the conference, there were also workshops on commutative algebra, algebraic geometry and representation theory, finite dimensional algebras, algebraic groups and lie theory, and quantum groups and hall algebras. An introduction to the mathematical structure of quantum mechanics. Hopf algebras, quantum groups and topological field theory. The text covers the material presented for a graduatelevel course on quantum groups at harvard university. Unrolled quantum groups are certain hopf algebras that were introduced in. We shall discuss generic extension monoids associated with finite dimensional basic hereditary algebras of finite or cyclic type and related applications to ringelhall algebras, and hence, to. The representation theory of infinite dimensional groups is an important tool for studying conformal field theory, problems in. Representations of finite dimensional algebras and related. We give a formulation in the framework of vertex algebras. In the past decade, there has been a sudden and vigorous development in a number of research areas in mathematics and mathematical physics, such as theory of operator algebras, knot theory, theory of manifolds, infinite dimensional lie algebras and quantum groups as a new topics, etc. Poisson algebras and quantization, poissonlie groups, coboundary lie bialgebras, drinfelds double construction, belavindrinfeld classification, infinite dimensional lie bialgebras, hopf algebras, quantized universal enveloping algebras. Quantum groups representations of the algebra u sl2, qorthogonal polynomials and invariants of links 285 a. Infinite dimensional algebras in chernsimons quantum. This construction builds algebras in a novel way by producing infinite generating sets via a process of mutation rather than the classical approach using generators and relations.
Bombay lectures on highest weight representations if infinite dimensional lie algebras subject. Infinite dimensional representations of quantum algebras. Pdf infinitedimensional groups and their representations. Pdf infinitedimensional lie groups and algebras in. An introduction peter woit department of mathematics, columbia university. Johnny t ottesen the representation theory of infinite dimensional groups is an important tool for studying conformal field theory, problems in statistical mechanics, and string theory. Infinitedimensional representations of quantum algebras. The book is far from elementary and is suitable for researchers and graduate students with a good knowledge of intersection theory and geometric invariant theory. Pdf dual pairs and infinite dimensional lie algebras. The table of contents for the full book pdf is as follows.
Representations of the algebra u q sl2, qorthogonal polynomials and invariants of links. This letter could not influence you to be smarter, however the book infinite dimensional groups and algebras in quantum physics lecture notes in physics monographs, by johnny t. Pdf ebook infinite dimensional groups and algebras in quantum physics lecture notes in physics monographs, by johnny t. Infinite dimensional groups and algebras in quantum. Buy infinite dimensional groups and algebras in quantum physics lecture notes in physics monographs on free shipping on qualified orders. Pdf infinite dimensional algebras and their applications. If g is a finitedimensional lie algebra acting on v possibly infinitedimensional by nilpotent endomorphisms, and v 0, then there exists a. Other readers will always be interested in your opinion of the books youve read. We discuss applications in quantum field theory and relativity gravity. Infinite dimensional lie algebras and groups advanced. Soliton equations infinite super grassmannians and super pliicker. The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations.
The interplay between finite dimensional algebras and lie theory dates back many years. We construct and study various dual pairs between finite dimensional classical lie groups and infinite dimensional lie algebras in some fock representations. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Starting with this knowledge, it surveys most of the known results and theories of the hilbert schemes of points, implying infinite dimensional lie algebras and their actions. For a finite dimensional semisimple lie algebra and a root of unity, lusztig defined an infinite dimensional quantum group of divided powers. Vertex algebras and quantum groups birs banff international. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. From purely mathematical point of view, affine lie algebras are interesting because their representation theory, like.
Proceedings of the infinite dimensional lie algebras and groups, luminy, marseille, france, 4 8 july 1988. The main algebraic approach to cluster algebras relies on representations of finite dimensional algebras and. This includes diffeomorphism groups and their natural subgroups like volumepreserving and symplectic transformations, as well as gauge groups and loop groups. It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. Bombay lectures on highest weight representations of infinite dimensional lie algebras. Infinite dimensional algebras in chernsimons quantum mechanics. This picture is different for infinitedimensional groups, and how bad it becomes. Quantum groups and algebras are of great importance for different branches of physics and mathematics. Infinite dimensional groups and algebras in quantum physics.
Four dimensional conformal field theory models withrational correlation functions n m nikolov, ya s stanev and i t todorovstring partition functions, hilbert schemes and affine lie algebra representations on homology groups loriano bonora, andrey bytsenko and emilio elizaldeconstraint quantization of a. An infinitedimensional hopf algebra over the field of rational. Quantum integrable systems and infinite dimensional algebras. A stabilization property derived from the multiplication formula permits the definition of an infinite dimensional algebra as a limit of all quantum schur algebras. Pdf quantum function algebras from finite dimensional.
Finite dimensional algebras, quantum groups and finite. Infinite dimensional algebras and quantum integrable. We show that h is not only a finite dimensional algebra but also a non co commutative hopf algebra, hence a finite dimensional quantum group. Character groups of hopf algebras as infinitedimensional. These include drinfeldjimbo type quantum groups which are quasitriangular hopf algebras, compact matrix quantum groups which are structures on unital separable c algebras, and bicrossproduct quantum groups. This volume presents the invited lectures of the workshop infinite dimensional algebras and quantum integrable systems held in july 2003 at the university of algarve, faro, portugal, as a satellite workshop of the xiv. In mathematics, an affine lie algebra is an infinite dimensional lie algebra that is constructed in a canonical fashion out of a finite dimensional simple lie algebra. Using the ideas of classical representation theory and basic facts of functional analysis, the author constructs the spin representations of the infinitesimal orthogonal group and the metaplectic representation of an infinite. Dennis gaitsgory harvard university metaplectic whittaker sheaves and quantum groups in this talk we will explain the construction of several versions of the fle, and equivalence between metaplectic whittaker sheaves and quantum groups for a group g and modules for the langlandsdual quantum group. We describe how to find quantum determinants and antipode formulas from braided vector spaces using the frtconstruction and finite dimensional nichols algebras.
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