Relaxing rain and thunder sounds, fall asleep faster, beat insomnia, sleep music, relaxation sounds duration. The real and imaginary roots 25 for kac moody algebras were introduced by kac 20. The second is the highest weight representations of the lie algebra g of infinite matrices, along. 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. 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. On complete reducibility for infinitedimensional lie algebras. Constructing groups associated to infinitedimensional lie. In the finite dimensional case this problem was solved by e. These algebras form a generalization of finite dimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. 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. 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. Highest weight modules over the lie algebra ga pages 103118.
In this thesis we consider the problem of classifying the automorphisms oforder two of an affine lie algebra. This is a lemma on pg 10 of kac s infinite dimensional lie algebras. Contents preface vii 1 preliminaries on afflne lie algebras 1 1. Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181. 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 classification of lnvolutive automorphisms of an affine.
Infinite dimensional lie algebras and superalgebras. The third is the unitary highest weight representations of the current affine kac moody algebras. Affine kacmoody algebras a natural problem is to generalize the theory of. 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. 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.
Infinite dimensional lie algebras memorial university. 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. Certain rank two subsystems of kac moody root systems. Part ii deals mainly with the representations and applications of lie algebras and contains many cross references to part i. Semi infinite homology of lie, kac moody and virasoro algebras in russian, uspekhi mat. The final classification of simple finite dimensional lie superalgebras is as. Integrable representations and the weyl group of a kac moody algebra. For many articles on kac moody algebras and related topics, one can refer sthanumoorthy and misra 240. Recently, a satisfactory theory, similar to lie s theory, has been developed on the connection between lie supergroups and lie superalgebras 5.
Symmetries and innite dimensional lie algebras martin schlichenmaier martin. Simultaneously, zgraded lie algebras were being studied in moscow where i. Some subalgebras of indefinite type kacmoody lie algebras. 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. Kazhdan, structure of representations with highest weight of infinite dimensional lie algebras, advances in math. Jason stephenson sleep meditation music 6,781,716 views. Infinite dimensional lie algebras and groups advanced. The material covered ranges from basic definitions of lie groups to the classification of finite dimensional representations of semisimple lie algebras. Infinitedimensional lie algebras harvard university. The proposed project is set in pure mathematics within the areas of infinite dimensional lie theory and geometric group theory. Kantor introduced and studied a general class of lie algebras including what eventually became known as kacmoody algebras. Infinite dimensional lie algebras an introduction victor g. Infinite dimensional lie algebras occurring in algebraic.
Shapovalov, on bilinear form on the universal enveloping algebra of a complex semisimple lie algebra,funkts. After this it only remains to reconstruct the lie superalgebra g with filtration from the zgraded lie superalgebra gr g. It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. Modular invariant representations of infinitedimensional lie. Modular invariant representations of infinite dimensional lie algebras and superalgebras, proc.
While all the simple finitedimensional lie algebras belong to the considered class of. Hook young diagrams, combinatorics and representations of lie superalgebras berele, a. 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. For the case of the infinite dimensional lie algebra, it leads to an infinite number of conserved classical quantities which generate hierarchy equations. 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. Use features like bookmarks, note taking and highlighting while reading infinitedimensional lie algebras. Infinitedimensional lie algebras, theta functions and modular forms author links open overlay panel victor g kac dale h peterson. There are lots of applications of them in for example, repre sentation theory. It will be shown that general polynomials of the kacmoody field can generate either a new infinite. 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. The higher kacmoody analogs of the exceptional semisimple lie algebras e7, e7, e8 are. A survey of the theory of kac polynomials for quivers and for curves.
Highest weight representations of infinite dimensional lie. An important subclass of infinite dimensional lie algebras among kac moody algebras is the socalled afline lie algebras. This survey article is written based on the authors seminar talks on nongraded infinite dimensional simple lie algebras. Continuing on, the main subjects of the book are the structure real and imaginary root systems of and the. Lectures on infinitedimensional lie algebra minoru wakimoto. Sthanumoorthy, in introduction to finite and infinite dimensional lie super algebras, 2016. An introduction progress in mathematics on free shipping on qualified orders. We specialize to integrable representations of kacmoody algebras and free lie algebras. We prove a character formula for a large class of highest weight representations l. In this paper, we launch a program to describe and classify modular invariant representations of infinitedimensional lie algebras and superalgebras. Solitons and infinite dimensional lie algebras by michio jimbo and tetsuji miwa contents introduction 1. Case for support unitary forms of kacmoody algebras and. This still gave rise to a lie algebra, but one which is now infinite dimensional.
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. 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. We study representations of the loop kac moody lie algebra g \otimes a, where g is any kac moody algebra and a is a ring of laurent polynomials in n commuting variables. Buy infinite dimensional lie algebras 3 by kac isbn. Infinitedimensional lie groups and algebras in mathematical. Infinite dimensional groups with applications victor kac.
Mathematical sciences research institute publications, vol 4. Its goal is to contribute to the structure theory of unitary forms i. 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. Introduction to finite and infinite dimensional lie super.
First let us explain the natural geometric construction of a. A second application is in the realm of soliton equations and their infinite dimensional symmetry groups and 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. Infinite dimensional lie algebras and groups by victor g kac. 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.
Infinitedimensional lie algebras kindle edition by kac, victor g download it once and read it on your kindle device, pc, phones or tablets. The book is concerned with kac moody algebras, a particular class of infinite dimensional lie algebras, and their representations. Nongraded infinite dimensional lie algebras appeared naturally in the theory of hamiltonian operators, the theory of vertex algebras and their multivariable analogues. The aim of this course is to inform students about theories of lie algebras, vertex algebras and other structures related to these theories. Kac massachusettsinstitute oftechnology, usa ashokk. In this paper we study the complete reducibility of representations of infinitedimensional lie algebras from the perspective of representation theory of vertex algebras. This is a joint work with thomas creutzig and andrew linshaw. When we specialize g to be a finite dimensional or affine lie algebra we obtain modules for toroidal lie algebras. Thence we can build a real formofthewittalgebraas x.
On complete reducibility for infinite dimensional lie algebras. In particular, we study representations with finite dimensional weight spaces and their graded versions. Everyday low prices and free delivery on eligible orders. 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. Infinite dimensional lie algebras and superalgebras youtube. Classification and construction of semisimple lie algebras.
From purely mathematical point of view, affine lie algebras are interesting because their representation theory, like. Infinitedimensional lie algebras, theta functions and. Global representations of the diffeomorphism groups of the circle. 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. Representation theory of superconformal algebras and the kac roanwakimoto conjecture. One of the key objects from nite dimensional theory is the cartan matrix. Affine kac moody lie algebras are one of the most interesting families of infinite dimensional lie algebras. Find a library or download libby an app by overdrive. Universal enveloping algebra and the casimir operator.
After recalling familiar concepts from classical theory we generalize to a much wider class of lie algebras known as kac moody algebras. Characters of highest weight modules over affine lie algebras. Other readers will always be interested in your opinion of the books youve read. This is an infinite dimensional lie algebra over c. Constructing groups from infinite dimensional lie algebras. The second is the highest weight representations of the lie algebra gl.
Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems. Raina tata institute offundamentalresearch,india natasha rozhkovskaya kansasstateuniversity, usa worldscientific new jersey london singapore beijing shanghai hong kong taipei chennai. The notion of imaginary roots has no counter part in the finite dimensional theory. A survey on invariant cones in infinite dimensional lie algebras. The witt algebra was a simple point of departure for infinite dimensional lie algebras.
Annihilators of verma modules for kacmoody lie algebras. Let l ct, t1 be the ring of laurent polynomials in t, and set lb l oc 0. It is a kac moody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. 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. Infinite dimensionallie algebras second edition victorg. The theoretical part largely deals with the representation theory of lie algebras with a. The three preceding examples all give the same lie algebra structure.
Xu, xiaoning and mu, bing, abstract and applied analysis, 20. 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. Integrating infinitedimensional lie algebras by a tannaka. The book is concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations. Representation theory of affine algebras, current groups and diff s 1. We show that the characters of all highest weight modules over an affine lie algebra with the highest weight away from the critical hyperplane are meromorphic functions in the positive half of the cartan subalgebra, their singularities being at most simple poles at zeros of real roots. The key constructional ingredients of our lie algebras. Characters of highest weight modules over affine lie. Bombay lectures on highest weight representations of infinite. Abstract this is the third, substantially revised edition of this important monograph by a giant in the field of mathematics. Annihilators of verma modules for kac moody lie algebras. In this paper we study the complete reducibility of representations of infinite dimensional lie algebras from the perspective of the representation theory of. Infinitedimensional lie algebras pdf free download epdf.