Affine kacmoody algebras a natural problem is to generalize the theory of. N sthanumoorthy lie superalgebras are a natural generalization of lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. The aim of this book is to establish relations between vertex operator algebras in mathematics and the string path integrals of physics. We formulate a notion of integrable modules over a symmetriz. 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 finite and infinite dimensional lie superalgebras. Highest weight representations of infinite dimensional lie. Arturo pianzola is the author of lie algebras with triangular decompositions, published by wiley. He is the codiscover of kacmoody algebra, a lie algebra, usually infinitedimensional, that can be defined through a generalized root system. Sthanumoorthy, in introduction to finite and infinite dimensional lie superalgebras, 2016. 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. These are the notes for a part iii course given in the university of cambridge in autumn 1998. 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. An introduction to affine kac moody lie algebras and groups. Concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations, this is the third revision of an important monograph.
Special issue on kac moody algebras and applications. Buy lie algebras with triangular decompositions on. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidlygrowing area of. These lectures were held at the graduate course on conformal field. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle.
Kacmoody lie algebras graded by kacmoody root systems. Uniqueness and presentation of kacmoody groups over fields. Kac moody groups, their flag varieties and representation. For many articles on kacmoody algebras and related topics, one can refer sthanumoorthy and misra 240. However, formatting rules can vary widely between applications and fields of interest or study. 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. 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. Wassermann, lecture notes on kac moody and virasoro algebras. Kass, moody, patera, and slanskys affine lie algebras.
Kacmoody 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. 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. We extend, to general kacmoody lie algebras, the notion of cadmissible pair as introduced by h.
This simplicial complex has the structure of a tits building, but is. 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. The notion of imaginary roots has no counter part in the finite. This lie algebra vdv contains the kacmoody algebra a as a subalgebra but is always far larger than a. Browse the amazon editors picks for the best books of 2019, featuring our. 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.
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. Lie algebras with triangular decompositions edition 1 by. This book is intended as a short concise overview of the theory of complex semisimple lie algebras. Special issue on kacmoody algebras and applications. Generalized kacmoody algebras are also sometimes called gkm algebras, borcherdskacmoody algebras, bkm algebras, or borcherds algebras. Wakimotos book makes a nice companion to kacs book and is filled. Uniqueness and presentation of kac moody groups over fields. Does anyone know a simple construction for affine kacmoody groups. Lie algebras have many varied applications, both in mathematics and mathematical physics.
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. Focuses on developing the theory of triangular decompositions and part of. Introduction to representation theory of kacmoody algebras. In joint work with alex feingold and walter freyn, we have constructed an action of hyperbolic kacmoody groups over. The book is concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations. Kacs book kac90l this selfcontained work treats the algebrogeometric and the topological aspects of kacmoody theory from scratch. 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. These algebras form a generalization of finitedimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems. Concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations, this is the third revision of an important monograph. 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. To reduce vdv to a smaller subalgebra, we will use the virasoro algebra. 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.
The third is the unitary highest weight representations of the current affine kac moody algebras. Bosonic construction of vertex operator paraalgebras from. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidlygrowing area of research. We shall not need more that basic notions on linear algebra and lie algebras. The material covered ranges from basic definitions of lie groups to the classification of finitedimensional representations of semisimple lie 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 kacmoody algebras. Moody, generalizing the finitedimensional semisimple lie. Lie algebras of finite and affine type cambridge studies. We name an indecomposable symmetrizable generalized cartan matrix a and the corresponding kacmoody lie algebra g. The second is the highest weight representations of the lie algebra gl.
The full chevalley theory for kacmoody groups is currently being worked out. Kacmoody groups, their flag varieties and representation theory. In particular, when lo is an affine kacmoody lie algebra and v is the basic representation of lo, we obtain a series of lorentzian kacmoody lie algebras e. The real and imaginary roots 25 for kacmoody algebras were introduced by kac 20. This lie algebra vdv contains the kac moody algebra a as a subalgebra but is always far larger than a. 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. Affine kac moody algebras a natural problem is to generalize the theory of.
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. This book is an introduction to a rapidly growing subject of modern mathematics, the. Kac s book kac 90l this selfcontained work treats the algebrogeometric and the topological aspects of kac moody theory from scratch. Introduction to kacmoody algebras world scientific publishing co. Browse the amazon editors picks for the best books of 2019, featuring our favorite.
Kac moody lie algebras 9 were introduced in the mid1960s independently by v. Kacmoody lie algebras where simultaneously introduced by kac and moody in the 1960s. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. What is an intuitive explanation of kacmoody algebras. The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations. Vertex algebras, kacmoody algebras, and the monster. I wonder if any one has calculated the quantum dimensions of those kac moody primary fields. The finite type indecomposable kacmoody algebras are precisely the. A systematic study of kacmoody algebras was started independently by v. The higher kacmoody analogs of the exceptional semisimple lie algebras e7, e7, e8 are. Kac moody groups and cluster algebras christof geiss, bernard leclerc, and jan schroer abstract.
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 book starts with an outline of the classical lie theory, used to set the scene. We extend, to general kac moody lie algebras, the notion of cadmissible pair as introduced by h. Introduction to finite and infinite dimensional lie super. The idea is to study the structure of each homogeneous space ln as a module over the affine kacmoody lie algebra lo. Kacmoody algebras and kacmoody groups, being natural extensions of classical lie algebras and linear algebraic groups, have been of constant interest. Kacs book kac90l this selfcontained work treats the algebrogeometric and the. The heart of the book is part iii, which develops an intuitive approach to the construction and fundamental properties of kacmoody groups.
In mathematics, a kacmoody algebra is a lie algebra, usually infinite dimensional, that can be. There is a book by kumar kacmoody groups, their flag varieties, and representation theory that does the construction for the. 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. 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. Throughout this chapter, all lie algebras are finite dimensional unless otherwise stated. Kacmoody groups, their flag varieties and representation theory by shrawan kumar, 9780817642273, available at book depository with free delivery worldwide. String path integral realization of vertex operator. 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. It serves as a reference book for physicists with commentary notes and reprints. 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.
Robert vaughan moody, oc frsc is a canadian mathematician. This volume is the proceedings of the ramanujan international symposium on kac moody lie algebras and their applications. Focuses on developing the theory of triangular decompositions and part of the kac moody theory not specific to the affine. This is the third, substantially revised edition of this important monograph. These lectures are intented to be a natural following of lectures on classical reprensentation theory. 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. A detailed treatment of the lie algebra aspect of the theory can be found in v. A clear account of kacmoody algebras by one of the founders. Moody, and subsequently many results of the theory of finitedimensional semisimple lie algebras have been carried over to kacmoody algebras. Kacmoody groups and cluster algebras christof geiss, bernard leclerc, and jan schroer abstract.
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. Affine kac moody algebras are natural generalizations of finitedimensional simple lie algebras, and they have many important applications. In these lectures we will start from the beginning the theory of lie algebras and their representations. He is the codiscover of kac moody algebra, a lie algebra, usually infinitedimensional, that can be defined through a generalized root system. Part of the progress in mathematics book series pm, volume 204. The cft of the sun level k kac moody current algebra has many kac moody primary fields.
Part ii provides a selfcontained introduction to kacmoody algebras. Kacmoody lie algebras, spectral sequences, and the witt. Automorphic forms and lie superalgebras urmie ray springer. We nish by discussing the structure and representation theory of reductive algebraic groups. How to compute irreducible representation of lie algebra in the framework of bbd. A lecture on kacmoody lie algebras of the arithmetic type viacheslavv. 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. Wassermann, lecture notes on kacmoody and virasoro algebras.
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