Kacmoody groups, their flag varieties and representation theory. This book is intended as a short concise overview of the theory of complex semisimple lie algebras. Let q be a nite quiver without oriented cycles, let be the associated preprojective algebra, let g be the associated kacmoody lie algebra with weyl group w, and let n be the positive part of g. Q with lie algebra product on v, but it is a lie algebra product on vdv, where dv is the image of v under a certain derivation d. This lie algebra vdv contains the kac moody algebra a as a subalgebra but is always far larger than a. Kac moody groups, their flag varieties and representation. There is also a section that constructs a onedimensional extension of the loop group by loop rotation, which is a fairly transparent definition. The finite type indecomposable kacmoody algebras are precisely the.
Bosonic construction of vertex operator paraalgebras from. Browse the amazon editors picks for the best books of 2019, featuring our favorite. This lie algebra vdv contains the kacmoody algebra a as a subalgebra but is always far larger than a. Part ii provides a selfcontained introduction to kacmoody algebras.
In joint work with alex feingold and walter freyn, we have constructed an action of hyperbolic kacmoody groups over. The material covered ranges from basic definitions of lie groups to the classification of finitedimensional representations of semisimple lie algebras. Focuses on developing the theory of triangular decompositions and part of. Wassermann, lecture notes on kacmoody and virasoro algebras. String path integral realization of vertex operator. The aim of this book is to establish relations between vertex operator algebras in mathematics and the string path integrals of physics. Lie algebras with triangular decompositions edition 1 by. Kacmoody groups, their flag varieties and representation theory by shrawan kumar, 9780817642273, available at book depository with free delivery worldwide. N sthanumoorthy lie superalgebras are a natural generalization of lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. The heart of the book is part iii, which develops an intuitive approach to the construction and fundamental properties of kacmoody groups. General properties of lie algebras, jordanchevalley decomposition, semisimple lie algebras, classification of complex semisimple lie algebras, cartan subalgebras, classification of connected coxeter graphs and complex semisimple lie. The book is concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations.
Introduction to kacmoody algebras world scientific publishing co. The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations. Let q be a nite quiver without oriented cycles, let be the associated preprojective algebra, let g be the associated kac moody lie algebra with weyl group w, and let n be the positive part of g. Robert vaughan moody, oc frsc is a canadian mathematician. Kacs book kac90l this selfcontained work treats the algebrogeometric and the. The notion of imaginary roots has no counter part in the finite. We shall not need more that basic notions on linear algebra and lie algebras. Kacmoody groups and cluster algebras christof geiss, bernard leclerc, and jan schroer abstract. Highest weight representations of infinite dimensional lie. These algebras form a generalization of finitedimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. There is a book by kumar kacmoody groups, their flag varieties, and representation theory that does the construction for the general kacmoody case, but i find the presentation dense. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle. A lecture on kacmoody lie algebras of the arithmetic type viacheslavv.
Special issue on kacmoody algebras and applications. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidlygrowing area of. This book is an introduction to a rapidly growing subject of modern mathematics, the. Part of the progress in mathematics book series pm, volume 204. Arturo pianzola is the author of lie algebras with triangular decompositions, published by wiley. A clear account of kacmoody algebras by one of the founders. The real and imaginary roots 25 for kacmoody algebras were introduced by kac 20. Buy bosonic construction of vertex operator paraalgebras from symplectic affine kac moody algebras memoirs of the american mathematical society on free shipping on qualified orders. He is the codiscover of kacmoody algebra, a lie algebra, usually infinitedimensional, that can be defined through a generalized root system.
The book starts with an outline of the classical lie theory, used to set the scene. Jun 29, 2010 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. Kac moody groups and cluster algebras christof geiss, bernard leclerc, and jan schroer abstract. To reduce vdv to a smaller subalgebra, we will use the virasoro algebra. They contain an exposition of the representation theory of the lie algebras of compact matrix groups, affine kac moody algebras and the virasoro algebra from a unitary point of view. Introduction to representation theory of kacmoody algebras. What is an intuitive explanation of kacmoody algebras. The second is the highest weight representations of the lie algebra gl. These lectures were held at the graduate course on conformal field. Wassermann, lecture notes on kac moody and virasoro algebras.
We extend, to general kacmoody lie algebras, the notion of cadmissible pair as introduced by h. However, formatting rules can vary widely between applications and fields of interest or study. 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. How to compute irreducible representation of lie algebra in the framework of bbd. Browse the amazon editors picks for the best books of 2019, featuring our. A principal ingredient in the proof of the moonshine theorem, connecting the monster group to modular forms, is the infinite dimensional lie algebra of physical states of a chiral string on an orbifold of a 26 dimensional torus, called the monster lie algebra. This is the third, substantially revised edition of this important monograph. An introduction to affine kac moody lie algebras and groups. Uniqueness and presentation of kacmoody groups over fields. 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. In this book we shall study infinitedimensional lie algebras associated both to the group mapm, g of smooth mappings from m to g and to the group of di.
There is a book by kumar kacmoody groups, their flag varieties, and representation theory that does the construction for the. Sthanumoorthy, in introduction to finite and infinite dimensional lie superalgebras, 2016. We nish by discussing the structure and representation theory of reductive algebraic groups. Kac moody lie algebras 9 were introduced in the mid1960s independently by v. Generalized kacmoody algebras are also sometimes called gkm algebras, borcherdskacmoody algebras, bkm algebras, or borcherds algebras. Throughout this chapter, all lie algebras are finite dimensional unless otherwise stated. He is the codiscover of kac moody algebra, a lie algebra, usually infinitedimensional, that can be defined through a generalized root system.
For many articles on kacmoody algebras and related topics, one can refer sthanumoorthy and misra 240. 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. Well, you have the theory of coxeter groups that can be defined in two following ways see reflection groups and coxeter groups cambridge studies in advanced mathematics. 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. I wonder if any one has calculated the quantum dimensions of those kac moody primary fields. The idea is to study the structure of each homogeneous space ln as a module over the affine kacmoody lie algebra lo. Numerous and frequentlyupdated resource results are available from this search. We name an indecomposable symmetrizable generalized cartan matrix a and the corresponding kacmoody lie algebra g.
This volume is the proceedings of the ramanujan international symposium on kac moody lie algebras and their applications. Moody, generalizing the finitedimensional semisimple lie. In these lectures we will start from the beginning the theory of lie algebras and their representations. The third is the unitary highest weight representations of the current affine kac moody algebras. Kacmoody algebras and kacmoody groups, being natural extensions of classical lie algebras and linear algebraic groups, have been of constant interest. Affine kac moody algebras a natural problem is to generalize the theory of.
Buy lie algebras with triangular decompositions on. Uniqueness and presentation of kac moody groups over fields. In mathematics, a kacmoody algebra is a lie algebra, usually infinite dimensional, that can be. These lectures are intented to be a natural following of lectures on classical reprensentation theory. In mathematics, a kac moody 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. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidlygrowing area of research. Automorphic forms and lie superalgebras urmie ray springer. Kass, moody, patera, and slanskys affine lie algebras. Kacmoody lie algebras where simultaneously introduced by kac and moody in the 1960s.
We formulate a notion of integrable modules over a symmetriz. 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 kacmoody algebras. While there is a number of good books on kacmoody algebras infinite dimensional lie algebras by kac is already enough, it seems to me. The characters for a large family of irreducible highest weight modules over a symmetrizable kac moody lie superalgebra are then given in terms of kazhdanlusztig polynomials for the.
Affine kac moody algebras are natural generalizations of finitedimensional simple lie algebras, and they have many important applications. These are the notes for a part iii course given in the university of cambridge in autumn 1998. The full chevalley theory for kacmoody groups is currently being worked out. This simplicial complex has the structure of a tits building, but is. 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. Does anyone know a simple construction for affine kacmoody groups. In mathematics, especially in lie theory, e n is the kac moody algebra whose dynkin diagram is a bifurcating graph with three branches of length 1, 2 and k, with k n. Lie algebras have many varied applications, both in mathematics and mathematical physics. Introduction to finite and infinite dimensional lie super. Concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations, this is the third revision of an important monograph. Inspite of its small volume, this text is far from being of easy lecture, since it assumes the knowledge of some basic facts concerning lie algebras, as well as associative algebras. Lie algebras of finite and affine type cambridge studies.
Concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations, this is the third revision of an important monograph. A detailed treatment of the lie algebra aspect of the theory can be found in v. 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. Special issue on kac moody algebras and applications. 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. Kacs book kac90l this selfcontained work treats the algebrogeometric and the topological aspects of kacmoody theory from scratch. 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. Moody, and subsequently many results of the theory of finitedimensional semisimple lie algebras have been carried over to kacmoody algebras. Focuses on developing the theory of triangular decompositions and part of the kac moody theory not specific to the affine. This book provides a thorough but relaxed mathematical treatment of the subject, including both the cartankillingweyl theory of finite dimensional simple algebras and the more modern theory of kac moody algebras.
Kacmoody lie algebras, spectral sequences, and the witt. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Kacmoody lie algebras graded by kacmoody root systems. Introduction to finite and infinite dimensional lie superalgebras. Vertex algebras, kacmoody algebras, and the monster. Wakimotos book makes a nice companion to kacs book and is filled. Kacmoody groups, their flag varieties and representation. A systematic study of kacmoody algebras was started independently by v. We extend, to general kac moody lie algebras, the notion of cadmissible pair as introduced by h. Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems. The higher kacmoody analogs of the exceptional semisimple lie algebras e7, e7, e8 are.