Affine kacmoody algebras a natural problem is to generalize the theory of. Integrable representations and the weyl group of a kac moody algebra. Constructing groups from infinite dimensional lie algebras. The three preceding examples all give the same lie algebra structure. Introduction to finite and infinite dimensional lie super. Infinite dimensional lie algebras and superalgebras youtube. Constructing groups associated to infinitedimensional lie. Abstract this is the third, substantially revised edition of this important monograph by a giant in the field of mathematics. Infinite dimensional lie algebras third edition infinite dimensional lie algebras third editionvictor g. Find a library or download libby an app by overdrive. Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems. Symmetries and innite dimensional lie algebras martin schlichenmaier martin. Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181.
The second is the highest weight representations of the lie algebra g of infinite matrices, along. Infinite dimensional groups with applications victor kac. This is a lemma on pg 10 of kac s infinite dimensional lie algebras. For many articles on kac moody algebras and related topics, one can refer sthanumoorthy and misra 240. Infinite dimensional lie algebras an introduction victor g. An introduction progress in mathematics on free shipping on qualified orders. Lie superalgebras appear in 4 as lie algebras of certain generalized groups, nowadays called lie supergroups, whose function algebras are algebras with commuting and anticommuting variables. The key constructional ingredients of our lie algebras.
It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. 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. This is the long awaited followup to lie algebras, part i which covered a major part of the theory of kac moody algebras, stressing primarily their mathematical structure. The material covered ranges from basic definitions of lie groups to the classification of finite dimensional representations of semisimple lie algebras. These algebras form a generalization of finite dimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. Representation theory of superconformal algebras and the kac roanwakimoto conjecture. Modular invariant representations of infinite dimensional lie algebras and superalgebras, proc. Characters of highest weight modules over affine lie.
First, the book contains an exposition on the role of finite dimensional semisimple lie algebras and their representations in the standard and grand unified models of elementary particle physics. Infinite dimensional lie algebras and groups advanced. The witt algebra was a simple point of departure for infinite dimensional lie algebras. There are lots of applications of them in for example, repre sentation theory. A survey of the theory of kac polynomials for quivers and for curves. This is a joint work with thomas creutzig and andrew linshaw. Jason stephenson sleep meditation music 6,781,716 views. Bombay lectures on highest weight representations of infinite. It is a kac moody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. Continuing on, the main subjects of the book are the structure real and imaginary root systems of and the. Infinitedimensional lie algebras kindle edition by kac, victor g download it once and read it on your kindle device, pc, phones or tablets.
Everyday low prices and free delivery on eligible orders. Infinitedimensional lie groups and algebras in mathematical. After this it only remains to reconstruct the lie superalgebra g with filtration from the zgraded lie superalgebra gr g. We specialize to integrable representations of kacmoody algebras and free lie algebras. Representation theory of affine algebras, current groups and diff s 1.
The notion of imaginary roots has no counter part in the finite dimensional theory. Infinite dimensional lie algebras occurring in algebraic. Hook young diagrams, combinatorics and representations of lie superalgebras berele, a. Let l ct, t1 be the ring of laurent polynomials in t, and set lb l oc 0. Recently, a satisfactory theory, similar to lie s theory, has been developed on the connection between lie supergroups and lie superalgebras 5. The real and imaginary roots 25 for kac moody algebras were introduced by kac 20. Semi infinite homology of lie, kac moody and virasoro algebras in russian, uspekhi mat. The third is the unitary highest weight representations of the current affine kacmoody algebras. Nongraded infinite dimensional lie algebras appeared naturally in the theory of hamiltonian operators, the theory of vertex algebras and their multivariable analogues. The proposed project is set in pure mathematics within the areas of infinite dimensional lie theory and geometric group theory. First let us explain the natural geometric construction of a. Relaxing rain and thunder sounds, fall asleep faster, beat insomnia, sleep music, relaxation sounds duration. We prove the conjecture on the gko construction of the minimal series principal w algebras in ade types, which goes back to the work of kac and wakimoto in 1990.
Other readers will always be interested in your opinion of the books youve read. Lectures on infinitedimensional lie algebra minoru wakimoto. Affine kac moody lie algebras are one of the most interesting families of infinite dimensional lie algebras. In mathematics, a kac moody algebra named for victor kac and robert moody, who independently discovered them is a lie algebra, usually infinite dimensional, that can be defined by generators and relations through a generalized cartan matrix. Infinitedimensional lie algebras harvard university. The third is the unitary highest weight representations of the current affine kac moody algebras. While discussing all classes of finite and infinite dimensional lie algebras and lie superalgebras in terms of their different classes of root systems, the book focuses on kac moody algebras. Annihilators of verma modules for kac moody lie algebras. In particular, kang and kim 37 discussed many interesting applications of their dimension formula to the various classes of graded lie algebras such as free lie algebras, kac moody algebras, and. Buy infinite dimensional lie algebras 3 by kac isbn. Mathematical sciences research institute publications, vol 4. Infinite dimensional lie algebras and superalgebras. In particular, we describe the representationtheoretic meaning of kac polynomials in terms of hall algebras, and the geometric meaning of kac polynomials in relation to the geometry of moduli spaces of representations of quivers or vector bundles on smooth projective curves. An important subclass of infinite dimensional lie algebras among kac moody algebras is the socalled afline lie algebras.
This is the third, substantially revised edition of this important monograph. Kazhdan, structure of representations with highest weight of infinite dimensional lie algebras, advances in math. Infinite dimensionallie algebras second edition victorg. When we specialize g to be a finite dimensional or affine lie algebra we obtain modules for toroidal lie algebras. Lie algebras by shlomo sternberg the campbell baker hausdorff formula, sl2 and its representations, classical simple algebras, engelliecartanweyl, conjugacy of cartan subalgebras, simple finite dimensional algebras, cyclic highest weight modules, serres theorem, and more. Classification and construction of semisimple lie algebras. A classification of lnvolutive automorphisms of an affine. We will study the theory of semisimple and kacmoody lie algebras, and some other in. Inner ideals were completely classi ed for nite dimensional simple lie algebras and recently this classi cation was extended to the nitary simple lie algebras by lopez, garcia. Infinite dimensional lie algebras and groups by victor g kac. From purely mathematical point of view, affine lie algebras are interesting because their representation theory, like. In this thesis we consider the problem of classifying the automorphisms oforder two of an affine lie algebra. Modular invariant representations of infinitedimensional lie.
The book is concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations. Lie algebras by shlomo sternberg the campbell baker hausdorff formula, sl2 and its representations, classical simple algebras, engel lie cartanweyl, conjugacy of cartan subalgebras, simple finite dimensional algebras, cyclic highest weight modules, serres theorem, and more. A second application is in the realm of soliton equations and their infinite dimensional symmetry groups and algebras. Raina tata institute offundamentalresearch,india natasha rozhkovskaya kansasstateuniversity, usa worldscientific new jersey london singapore beijing shanghai hong kong taipei chennai. It will be shown that general polynomials of the kacmoody field can generate either a new infinite. The second is the highest weight representations of the lie algebra gl. 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. An analytic approach e rodriguezcarrington almost splitkforms of kac moody algebras g rousseau. Certain rank two subsystems of kac moody root systems.
Infinite dimensional lie algebras occurring in algebraic geometry spring 2015 content of the course this course is set up as an introduction to the theory of conformal blocks wesszuminowitten theory, whose ultimate goal is described below. Kac massachusettsinstitute oftechnology, usa ashokk. The book is concerned with kac moody algebras, a particular class of infinite dimensional lie algebras, and their representations. Part ii deals mainly with the representations and applications of lie algebras and contains many cross references to part i. The notion of kacmoody lie algebra is a generalization of that of semisimple lie algebra to infinite dimension of the underlying vector space definition examples. Simultaneously, zgraded lie algebras were being studied in moscow where i. On complete reducibility for infinitedimensional lie algebras. Global representations of the diffeomorphism groups of the circle. Some subalgebras of indefinite type kacmoody lie algebras.
The higher kacmoody analogs of the exceptional semisimple lie algebras e7, e7, e8 are. Infinitedimensional lie algebras, theta functions and modular forms author links open overlay panel victor g kac dale h peterson. In particular, we study representations with finite dimensional weight spaces and their graded versions. There is also a section that constructs a one dimensional extension of the loop group by loop rotation, which is a fairly transparent definition. Xu, xiaoning and mu, bing, abstract and applied analysis, 20. Its goal is to contribute to the structure theory of unitary forms i. Use features like bookmarks, note taking and highlighting while reading infinitedimensional lie algebras. Contents preface vii 1 preliminaries on afflne lie algebras 1 1. This survey article is written based on the authors seminar talks on nongraded infinite dimensional simple lie algebras.
This volume begins with an introduction to the structure of finite dimensional simple lie algebras, including the representation of \\widehat \mathfrak sl2, \mathbb c\, root systems, the cartan matrix, and a dynkin diagram of a finite dimensional simple lie algebra. This still gave rise to a lie algebra, but one which is now infinite dimensional. Infinitedimensional lie algebras, theta functions and. Highest weight modules over the lie algebra ga pages 103118. This is an infinite dimensional lie algebra over c.
After recalling familiar concepts from classical theory we generalize to a much wider class of lie algebras known as kac moody algebras. The final classification of simple finite dimensional lie superalgebras is as. In the finite dimensional case this problem was solved by e. Concerned with kac moody algebras, a particular class of infinite dimensional lie algebras, and their representations, this is the third revision of an important monograph. A survey on invariant cones in infinite dimensional lie algebras. The theoretical part largely deals with the representation theory of lie algebras with a. Sthanumoorthy, in introduction to finite and infinite dimensional lie super algebras, 2016. In this paper we study the complete reducibility of representations of infinite dimensional lie algebras from the perspective of the representation theory of. Kantor introduced and studied a general class of lie algebras including what eventually became known as kacmoody algebras. Jun 29, 2010 representations of loop kac moody lie algebras. It is based on courses given over a number of years at mit and in paris, and is sufficiently selfcontained and detailed to be used for graduate courses. On complete reducibility for infinite dimensional lie algebras.
Solitons and infinite dimensional lie algebras by michio jimbo and tetsuji miwa contents introduction 1. Irreducible characters of kacmoody lie superalgebras 3 while super duality in our general setting is established largely by the same strategy as in the case of. Characters of highest weight modules over affine lie algebras. In this paper, we launch a program to describe and classify modular invariant representations of infinitedimensional lie algebras and superalgebras. This volume records most of the talks given at the conference on infinite dimensional groups held at the mathematical sciences research institute at berkeley, california, may 10may 15, 1984, as a part of the special program on kac moody lie algebras. Infinitedimensional lie algebras pdf free download epdf.
While all the simple finitedimensional lie algebras belong to the considered class of. In this paper we study the complete reducibility of representations of infinitedimensional lie algebras from the perspective of representation theory of vertex algebras. In mathematics, an affine lie algebra is an infinitedimensional lie algebra that is constructed in a canonical fashion out of a finitedimensional simple lie algebra. Case for support unitary forms of kacmoody algebras and. Universal enveloping algebra and the casimir operator. There is a book by kumar kac moody groups, their flag varieties, and representation theory that does the construction for the general kac moody case, but i find the presentation dense. One of the key objects from nite dimensional theory is the cartan matrix. Highest weight representations of infinite dimensional lie. Annihilators of verma modules for kacmoody lie algebras.
Shapovalov, on bilinear form on the universal enveloping algebra of a complex semisimple lie algebra,funkts. For the case of the infinite dimensional lie algebra, it leads to an infinite number of conserved classical quantities which generate hierarchy equations. The aim of this course is to inform students about theories of lie algebras, vertex algebras and other structures related to these theories. Thence we can build a real formofthewittalgebraas x.