Kac moody algebra pdf book

Infinite dimensional lie algebras an introduction progress. This is the kac moody algebra with cartan matrix a. Affine kacmoody algebras a natural problem is to generalize the theory of. Uniqueness and presentation of kacmoody groups over fields. They contain an exposition of the representation theory of the lie algebras of compact matrix groups, affine kacmoody algebras and the virasoro algebra from a unitary point of view. Colored minuscule and dcomplete partially ordered sets encode information that can be used to construct many objects that arise in. This is the first monograph to exclusively treat kacmoody km groups, a standard tool in mathematics and mathematical physics. Universal envelopping algebras, levis theorem, serres theorem, kac moody lie algebra, the kostants form of the envelopping algebra and a beginning of a proof of the chevalleys theorem. Spherical hecke algebras for kacmoody groups over local. On the homology of the kacmoody groups and the cohomology of the 3connective covers of lie groups nishimura, osamu, journal of mathematics of kyoto university, 2002. We define the spherical hecke algebra for an untwisted affine kac moody group over a local nonarchimedian field. It serves as a reference book for physicists with commentary notes and reprints. Our main references are the original paper by tits tit87 and the book by rem y rem02.

Then s i is isomorphic to sl 2, and s i acts on ga via the adjoint action. Kacs book kac90l this selfcontained work treats the algebrogeometric and the topological aspects of kacmoody theory from scratch. We prove a generalization of the satake isomorphism for this algebra, relating it to integrable representations of the langlands dual affine kac moody group. Kacmoody groups, their flag varieties and representation. Linear algebra and group representations download ebook pdf. Special issue on kacmoody algebras and applications. We mention below two such constructions both due to beilinson and drinfeld. Km lie algebras were introduced in the midsixties independently by v. Working with fractions is a very important foundation to algebra. Now let a be an arbitrary symmetrizable irreducible cartan matrix. Kacs book kac90l this selfcontained work treats the algebrogeometric and the. Borcherds, trinity college, cambridge cb2 1tq, england.

The second additional application is given in this article by daboul, daboul and slodowy where a dynamical algebra of the full kepler problem was found to be a twisted kacmoody algebra. This volume is the proceedings of the ramanujan international symposium on kacmoody lie algebras and their applications. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidlygrowing area of research. Kacmoody groups over a field in this preliminary section we recall the basic notions and notations of kacmoody groups over a eld.

The theoretical part largely deals with the representation theory of lie algebras with a triangular decomposition, of which kacmoody algebras and the virasoro algebra are prime examples. Let m be the unique maximal ideal which intersects h trivially. Infinite dimensional lie algebras an introduction progress in. We shall not need more that basic notions on linear algebra and lie algebras.

Request pdf kacmoody and virasoro algebras these are the notes for a part iii course given in the university of cambridge in autumn 1998. Integrable modules for each i 1n, let s i be the lie subalgebra with vector space basis fe i. A generalized cartan matrix a and the corresponding kac moody lie algebra g. The book begins by establishing a practical basis in lie algebra, including discussions of structure lie, loop, and virasor groups, quantum tori and kacmoody algebras, and gradation. We define the spherical hecke algebra for an untwisted affine kacmoody group over a local nonarchimedian field. A have the arithmetic type if a is symmetrizable indecomposable and for the corresponding canonical symmetric bilinear form on the. Kacmoody algebras and their representations xiaoping xu on. We introduce generalizations of kacmoody 2categories in which the quiver hecke algebras of khovanov, lauda and rouquier are replaced by the quiver hecke superalgebras of kang, kashiwara and tsuchioka.

Kac moody algebra an n nmatrix is called a generalized cartan matrix if for i. This is an introduction to the basic ideas and to a few further selected topics in conformal quantum eld theory and in the theory of kacmoody algebras. The inhouse pdf rendering service has been withdrawn. 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. Characterization and classification of minuscule kacmoody. A hyperbolic kacmoody algebra and the theory of siegel modular forms of genus 2. College of arts and sciences, department of mathematics. These are the notes for a part iii course given in the university of cambridge in autumn 1998. In mathematics, a generalized kacmoody algebra is a lie algebra that is similar to a kacmoody algebra, except that it is allowed to have imaginary simple roots. A systematic study of kacmoody algebras was started independently by v. Superstrings, pbranes and mtheory theories of everything. Thisbooksemphasisonmotivationanddevelopment,anditsavailability, makeitwidelyusedforselfstudy.

Lies third theorem, the existence of a lie group with given. After setting up the general framework of highest weight representations, the book continues to treat topics as the casimir operator and the weylkac. Kacmoody lie algebras of the arithmetic type 3 for any 1. Wassermann, lecture notes on kacmoody and virasoro algebras. This site is like a library, use search box in the widget to get ebook that you want. Download introductionto kac moody algebras ebook pdf or read online books in pdf, epub, and mobi format. A kacmoody group functor depends on a set of data d consisting of a n n. Moody, and subsequently many results of the theory of finitedimensional semisimple lie algebras have been carried over to kacmoody algebras. What is a good reference for a fast approach to construct affine kacmoody algebras from finitedimensional simple lie algebras. These lectures are intented to be a natural following of lectures on classical reprensentation theory.

It then offers a detailed discussion of prolongation structure and its representation theory, the orbit approachfor both finite and infinite dimension lie algebra. This book is an introduction to a rapidly growing subject of modern mathematics, the kac moody algebra, which was introduced by v kac and r moody simultanously and independently in 1968. Reduce, add, subtract, multiply, and divide with fractions. Click download or read online button to get linear algebra and group representations book now. Introduction to kac moody algebra download ebook pdf. In mathematics, a kacmoody algebra named for victor kac and robert moody, who independently discovered them is a lie algebra, usually infinitedimensional, that can be defined by generators and relations through a generalized cartan matrix. It is based on lectures and seminars given by professor kac over the last few years at mit.

The spherical hecke algebra for affine kacmoody groups i. This is a wikipedia book, a collection of wikipedia articles that can be easily saved, imported by an external electronic rendering. This is the first monograph to exclusively treat kac moody km groups, a standard tool in mathematics and mathematical physics. Quantizations of kacmoody algebras article in journal of pure and applied algebra 2184. The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations. Noncommutative poisson algebra structures on affine kac. Generalized kacmoody algebras are also sometimes called gkm algebras, borcherdskacmoody algebras, bkm algebras, or borcherds algebras. A survey of the theory of unitary representations of kacmoody algebras is given in. Kazhdan, representation of affine kacmoody groups over local and global fields. Why do physicists need unitary representation of kacmoody.

Linear algebra and group representations download ebook. Vertex algebras, kacmoody algebras, and the monster. These lectures were held at the graduate course on conformal field. Kacmoody lie algebras where simultaneously introduced by kac and moody in the 1960s. In order to study infinitedimensional lie algebras with root space decomposition as finitedimensional simple lie algebras. Moody, generalizing finitedimensional semisimple lie algebras. 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. Lie algebraic methods in integrable systems crc press book.

Kacmoody groups, their flag varieties and representation theory. This book is an introduction to a rapidly growing subject of modern mathematics, the kacmoody algebra, which was introduced by v kac and r moody simultanously and independently in 1968. Kacmoody algebra quantum anomaly lie algebra affine lie algebra string string theory. Beginning and intermediate algebra cabrillo college. I know that kacs book and many others do a very detailed and progressive construction, but i mean a understandable and direct realization as in the hong and kangs book about quantum groups. We prove a generalization of the satake isomorphism for this algebra, relating it to integrable representations of the langlands dual affine kacmoody group. Kacmoody algebras and kacmoody groups, being natural extensions of classical lie algebras and linear algebraic groups, have been of constant interest. This book is written at the level of a first course in calculus and linear algebra and is aimed at undergraduate and beginning graduate students in mathematics, computer science, and physics. The second additional application is given in this article by daboul, daboul and slodowy where a dynamical algebra of the full kepler problem was found to be a twisted kac moody algebra. Kac moody algebras and their representations xiaoping xu on. Kac s book kac 90l this selfcontained work treats the algebrogeometric and the topological aspects of kac moody theory from scratch. A detailed treatment of the lie algebra aspect of the theory can be found in v. Noncommutative poisson algebra structures on kacmoody algebras throughout this paper we let a aij.