Infinite dimensional groups with applications victor kac. Almost simultaneously in 1967, victor kac in the ussr and robert moody in canada developed what was to become kac moody algebra. Infinite dimensional lie algebras an introduction progress in. The book is concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations. Projective unitary representations of infinite dimensional lie groups janssens, bas and neeb, karlhermann, kyoto journal of mathematics, 2019. On complete reducibility for infinitedimensional lie algebras. They are characterized by the existence of a finite dimensional cartan subalgebra, a nondegenerate symmetric. Classification and construction of semisimple lie algebras.
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. The book is concerned with kac moody algebras, a particular class of infinite dimensional lie algebras, and their representations. A finitedimensional lie algebra is characterized, relative to a basis, by a set of numbers, called structure constants that specify the bracket operation. Affine kacmoody algebras a natural problem is to generalize the theory of. The three preceding examples all give the same lie algebra structure. Infinite dimensional lie algebras an introduction victor. Use features like bookmarks, note taking and highlighting while reading infinitedimensional lie algebras. Thence we can build a real formofthewittalgebraas x. Kantor introduced and studied a general class of lie algebras including what eventually became known as kacmoody algebras. Michor institut fur mathematik, universit at wien, strudlhofgasse 4, a1090 wien. Chapter 18 extensions of lie algebras pages 548 download pdf.
We study a class of possibly infinite dimensional lie algebras, called the quasisimple lie algebras qslas, and generalizing semisimple and affine kac moody lie algebras. Simultaneously, zgraded lie algebras were being studied in moscow where i. Victor kac, infinite dimensional lie algebras, cambridge university press 1990 collections of articles include. Universal enveloping algebra and the casimir operator. From this we deduce a construction of all levelone highestweight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra. Descargar infinitedimensional lie algebras en pdf libros. The table of contents for the full book pdf is as follows. Infinite dimensionallie algebras second edition victorg. Infinite dimensional lie algebras and groups advanced.
Integrable representations and the weyl group of a kac moody algebra. 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. Spin and wedge representations of infinitedimensional lie. Highest weight modules over the lie algebra ga pages 103118. 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. For a kacmoody group, this produces three dimensional chernsimons theory, while for the virasoro group the result is a new three dimensional topological. Finitedimensional lie algebra an overview sciencedirect.
Click download or read online button to get infinite dimensional lie algebras an introduction progress in mathematics book now. Lie theory, 7,1 1997, 6199 regular infinite dimensional lie groups andreas kriegl peter w. Kacmoody lie algebras, spectral sequences, and the witt. Infinite dimensional lie algebras third edition infinite dimensional lie algebras third editionvictor g. In this paper we construct two kinds of infinitedimensional 3 lie algebras from a given commutative associative algebra, and show that they are all canonical nambu 3 lie algebras. The real and imaginary roots 25 for kac moody algebras were introduced by kac 20. Modules for kac moody algebras s kumar frobenius action on the bcohomology o mathieu certain rank two subsystems of kac moody root systems j morita lie groups associated to kac moody lie algebras. Jul 04, 2007 contragredient lie algebras and lie algebras associated with a standard pentad sasano, nagatoshi, tsukuba journal of mathematics, 2018. Laplace operators of infinitedimensional lie algebras and. In section 9 and section 10 we treat two more typical examples of soliton equations, the 2 dimensional toda lattice. First let us explain the natural geometric construction of a. Modular invariant representations of infinitedimensional. Included are the infinite dimensional orthogonal or symplectic lie algebras b, c, d and the kac moody lie algebras of euclidean type.
On complete reducibility for infinitedimensional lie algebras article in advances in mathematics 2262. Kazhdan, structure of representations with highest weight of infinite dimensional lie algebras, advances in math. Most relevant for the first part of these notes are chapterss 7 and 14 of kacs book. Each chapter begins with a motivating discussion and ends.
Each copy of any part of this document must contain these terms of. Laplace operators of infinite dimensional lie algebras and theta functions. Contents preface vii 1 preliminaries on afflne lie algebras 1 1. 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. This is the third, substantially revised edition of this important monograph. Chapter 19 explicit construction of affine kac moody algebras.
We prove a character formula for a large class of highest weight representations l. The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations. Once that is accomplished, we will have a much deeper understanding of the structure of kac moody lie algebras and their connections with. While all the simple finitedimensional lie algebras belong to the considered class of. We suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4. Antony wassermann, kac moody and virasoro algebras, course notes 2011 hermann nicolai, infinite dimensional symmetries 2009 the standard textbook is. 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. This is a joint work with thomas creutzig and andrew linshaw. Cambridge core algebra infinitedimensional lie algebras by victor g. Gomez that any nondegenerate lie algebra is a subdirect product of strongly prime lie algebras, which allows us to reduce the proof to the strongly prime case, thus avoiding the di culty in the original. These algebras form a generalization of finite dimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such.
Finite and infinite dimensional lie algebras and applications in physics e. Pdf a survey on invariant cones in infinite dimensional. This is the long awaited followup to lie algebras, part i which covered a major part of the theory of kac moody algebras. We will also talk some about kac moody algebras, a mild generalization of lie algebras, which allows them to be infinite dimensional, and along the way the heisenberg algebra of infinitely many things, and the virasoro algebra. Other readers will always be interested in your opinion of the books youve read. A particular wellunderstood class of in nite dimensional kac moody algebras, the a ne lie algebras, have found numerous applications throughout mathematics and physics. Infinitedimensional lie algebras pdf free download epdf. A hardyramanujan formula for lie algebras ritter, gordon, experimental mathematics, 2007. Infinitedimensional lie groups and algebras in mathematical.
Kac massachusettsinstitute oftechnology, usa ashokk. Read lectures on infinite dimensional lie algebra by minoru wakimoto for online ebook. An introduction progress in mathematics on free shipping on qualified orders. Infinitedimensional lie algebras, theta functions and. As is well known, one can express a character of a lie. Kitchloo submitted to the department of mathematics on may 5, 1998 in partial fulfillment of the requirements for the degree of doctor of philosophy abstract the class of complex semisimple lie algebras can be extended to include infinite dimensional lie algebras known as kac moody lie algebras. Infinitedimensional lie algebras kindle edition by kac, victor g download it once and read it on your kindle device, pc, phones or tablets. Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems. Modular invariant representations of infinitedimensional lie. The witt algebra was a simple point of departure for infinite dimensional lie algebras. Developments and trends in infinitedimensional lie theory. 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. This still gave rise to a lie algebra, but one which is now infinite dimensional.
A procedure for constructing topological actions from centrally extended lie groups is introduced. Kac 1 department of mathematics, massachusetts institute of technology, cambridge, ma 029. In this paper, we launch a program to describe and classify modular invariant representations of infinitedimensional lie algebras and superalgebras. The book is concerned with kacmoody algebras, a particular class of infinite dimensional lie algebras, and their representations. This includes diffeomorphism groups and their natural subgroups like volumepreserving and symplectic transformations, as well as gauge groups and loop groups. Let l ct, t1 be the ring of laurent polynomials in t, and set lb l oc 0.
Sthanumoorthy, in introduction to finite and infinite dimensional lie super algebras, 2016. Lectures on infinitedimensional lie algebra minoru wakimoto. This is an infinite dimensional lie algebra over c. Continuing on, the main subjects of the book are the structure real and imaginary root systems of and the. Raina tata institute offundamentalresearch,india natasha rozhkovskaya kansasstateuniversity, usa worldscientific new jersey london singapore beijing shanghai hong kong taipei chennai. On simplicity of certain infinite dimensional lie algebras. Thus if v 1, v n is a basis of a lie algebra v, we define the structure constants c k ij, i, j, k 1, n by expanding, in terms of this basis, the bracket of any two basis elements. His brother boris katz is a principal research scientist at mit. For many articles on kac moody algebras and related topics, one can refer sthanumoorthy and misra 240. Lie algebras constructed with lie modules and their positively and negatively graded modules sasano, nagatoshi, osaka journal of mathematics, 2017.
Lie algebras finite and infinite dimensional lie algebras and applications in physics. Chapter 19 explicit construction of affine kac moody algebras pages 4970 download pdf. Shapovalov, on bilinear form on the universal enveloping algebra of a complex semisimple lie algebra,funkts. Part a is mainly concerned with the structure and representation theory of infinite dimensional lie algebras and contains articles on the structure of directlimit lie algebras, extended affine lie algebras and loop algebras, as well as representations of loop algebras and kac moody superalgebras. 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. An analytic approach e rodriguezcarrington almost splitkforms of kac moody algebras g rousseau. Mathematical methods for the natural and engineering sciences.
Laplace operators of infinite dimensional lie algebras and theta functions victor g. Kac and moody noticed that if wilhelm killings conditions were relaxed, it was still possible to associate to the cartan matrix a lie algebra which, necessarily, would be infinite dimensional. Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181. Infinite dimensional lie algebras an introduction progress. 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. Pdf a survey on invariant cones in infinite dimensional lie. 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.