Communicated by walter feit, december, 1985 abstract it is known that the adjoint representation of any kacmoody. Shari a prevost we present a new proof of the identities needed to exhibit an explicit boldzbasis for the universal enveloping algebra associated to. Affine lie algebras, weight multiplicities, and branching. Kacmoody groups, their flag varieties and representation. The idea is to study the structure of each homogeneous space ln as a module over the affine kacmoody lie algebra lo. A study on the structure of indefinite quasiaffine kac. Kacmoody lie algebras where simultaneously introduced by kac and moody in the 1960s.
Extended affine lie algebras and their root systems. 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. It is clear that this book is a valuable reference for all those interested in flag varieties and representation theory in the semisimple or kacmoody case. Borcherds, trinity college, cambridge cb2 1tq, england. This is the third, substantially revised edition of this important monograph. In this volume, this general idea can be seen at work from several points of view. Different definitions of highest weight representation for. The following references discuss aspects of the kacmoody exceptional geometry of supergravity theories. It is widely accepted in the mathematical community that the best way to study the representation theory of affine kacmoody algebras is by investigating the representation theory of the associated affine vertex and walgebras. We shall not need more that basic notions on linear algebra and lie algebras. Their infinite dimensional counter parts have recently been discovered which are affine kacmoody symmetric spaces. Does anyone know a simple construction for affine kacmoody groups. An indefinite matrix s gives rise to a kac moody algebra of indefinite type.
Affine lie algebra an overview sciencedirect topics. The reader is assumed to be familiar with the theory of semisimple lie algebras, e. A detailed treatment of the lie algebra aspect of the theory can be found in v. It serves as a reference book for physicists with commentary notes and reprints.
This monograph provides a crystal clear exposition of the theory of finitedimensional simple lie algebras and a nice introduction to infinitedimensional kac moody algebras of affine type. Vertex algebras and integral bases for the enveloping algebras of affine lie algebras. These are the notes for a part iii course given in the university of cambridge in autumn 1998. Kac moody lie algebras are described by extended dynkin diagrams 4. We prove a generalization of the satake isomorphism for this algebra, relating it to integrable representations of the langlands dual affine kacmoody group. There is a book by kumar kacmoody groups, their flag varieties, and representation theory that does the construction for the. The authors make the first steps toward reformulating the theory of vertex algebras in a way that is suitable for algebraicgeometric applications. The finite dimensional simple lie algebras are examples of kacmoody algebras. For the theory of affine lie algebras the book by kac is an essential foundational source, but a possibly more readable textbook account of his ideas was given by roger carter in his 2005 cambridge university press book lie algebras of finite and affine type.
We define the spherical hecke algebra for an untwisted affine kac moody group over a local nonarchimedian field. From purely mathematical point of view, affine lie algebras are interesting because their representation theory, like. Since the diagonal entries of c and s are positive, s cannot be negative definite or negative semidefinite. Affine kacmoody symmetric spaces related with asub 1.
In this chapter we study central extensions of free abelian and finite abelian groups. In this paper we have algebraically explicitly computed the affine kacmoody symmetric spaces associated with affine kacmoody algebras. Dieudonnp received june 11, 1986 in this thesis we classify the conjugacy classes of involutions in aut g, where g is an afline kacmoody lie algebra. We nish by discussing the structure and representation theory of reductive algebraic groups.
The emphasis is on the study of the kacmoody groups 9 and their flag varieties xy, including their detailed construction, and. A ne lie algebras kevin wray january 16, 2008 abstract in these lectures the untwisted a ne lie algebras will be constructed. Observe that thanks to the existence of a nondegenerate symmetric bilinear form b on the target algebra, loop algebras always have a central extension given by the cocycle. A positive semidefinite matrix s gives rise to an infinitedimensional kac moody algebra of affine type, or an affine lie algebra. The book is concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations. The second half introduces the theory of kacmoody algebras, concentrating particularly on those of affine type. Shari a prevost we present a new proof of the identities needed to exhibit an explicit boldzbasis for the universal enveloping algebra associated to an affine lie algebra. 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. It is a kac moody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. Lie algebras of finite and affine type roger carter. Daniel beltita, institute of mathematics, romanian academy, siam.
They should be viewed as tangent lie algebras to the loop groups, with a correction term which is sometimes related to quantizationquantum anomaly these affine lie algebras appear in quantum field theory as the current algebras in the wzw model as well as in its chiral halfs. A ne lie algebras university of california, berkeley. Dieudonnp received june 11, 1986 in this thesis we classify the conjugacy classes of involutions in aut g, where g is an afline kac moody lie algebra. Lie algebras of finite and affine type book, 2005 worldcat. It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. 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. Humphreys introduction to lie algebras and representation theory 1. Provides a foundation for the study of an important class of lie algebras that generalizes the class of affine kacmoody lie algebras. Lie algebras of finite and affine type by carter, roger. 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 kacmoody algebras.
Local geometric langlands correspondence and affine kac. Kacmoody lie algebras are described by extended dynkin diagrams 4. Also, an excellent course on lie algebras could be constructed starting from the book. Affine, vertex and walgebras drazen adamovic springer. Finite order automorphisms and real forms of affine kac moody algebras in the smooth and algebraic category memoirs of the american mathematical society, number 1030 9780821869185. This paper deals with the problem of conjugacy of cartan. Vertex algebras and algebraic curves edward frenkel. Vertex algebras and integral bases for the enveloping. A positive semidefinite matrix s gives rise to an infinitedimensional kacmoody algebra of affine type, or an affine lie algebra. Lie algebras of finite and affine type roger carter bok. We present an explicit character formula for the irreducible highest weight representations of the nontwisted affine kacmoody lie algebra at the critical level which are integrable over the corresponding finitedimensional simple lie algebra. A brief account of borcherds algebras is also included.
The second half introduces the theory of kac moody algebras, concentrating particularly on those of affine type. The first half of the book deals with classification of the finite dimensional simple lie algebras and of their finite dimensional irreducible representations. Finite order automorphisms and real forms of affine kacmoody algebras in the smooth and algebraic category memoirs of the american mathematical society, number 1030 9780821869185. A classification of lnvolutive automorphisms of an affine kac. In fuchs book on affine lie algebras and quantum groups. Volume one contains a description of kacmoody lie algebras, and especially the affine algebras and their representations.
We extend, to general kacmoody lie algebras, the notion of cadmissible pair as introduced by h. The book will be useful to lie algebraists, high energy physicists, statistical mechanics, and number theorists. Though carter also tends to work by default just over the complex field, its. A systematic study of kacmoody algebras was started independently by v. A kacmoody algebra also kacmoody lie algebra is defined as follows.
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. We prove a generalization of the satake isomorphism for this algebra, relating it to integrable representations of the langlands dual affine kac moody group. Bosonic construction of vertex operator paraalgebras from. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship between vertex algebras and the geometry of moduli spaces of algebraic curves. 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. These extensions are required in order to modify the vertex operators which have provided representations of the affine algebra in chapters 3 and 4. We define the spherical hecke algebra for an untwisted affine kacmoody group over a local nonarchimedian field. Provides a foundation for the study of an important class of lie algebras that generalizes the class of affine kac moody lie algebras. We extend, to general kac moody lie algebras, the notion of cadmissible pair as introduced by h. Vertex algebras and algebraic curves edward frenkel, david. Vertex algebras, kacmoody algebras, and the monster.
They should be viewed as tangent lie algebras to the loop groups, with a correction term which is sometimes related to quantizationquantum anomaly. Moody independently introduced the lie algebras now known as kacmoody algebras. Lie algebras of finite and affine type cambridge studies. Affine kacmoody algebras a natural problem is to generalize the theory of. As far as applications this is the most important class. Kacs book kac90l this selfcontained work treats the algebrogeometric and the topological aspects of kacmoody theory from scratch. An appendix gives a summary of the basic properties of each lie algebra of finite and affine type. 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. These lectures are intented to be a natural following of lectures on classical reprensentation theory. The idea is to study the structure of each homogeneous space ln as a module over the affine kac moody lie algebra lo.
Some appendices recall useful results from other areas, so the work may be considered selfcontained, although some familiarity with semisimple lie algebras or algebraic groups is helpful. The affine kacmoody algebra equationis the universal central extension of the formal. In particular, when lo is an affine kac moody lie algebra and v is the basic representation of lo, we obtain a series of lorentzian kac moody lie algebras e. Qa article june 2007 with 36 reads how we measure reads. We present an explicit character formula for the irreducible highest weight representations of the nontwisted affine kac moody lie algebra at the critical level which are integrable over the corresponding finitedimensional simple lie algebra. Volume one contains a description of kac moody lie algebras, and especially the affine algebras and their representations. These lie algebras and related notions were first defined and studied by kac and moody. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. Lie algebras of finite and affine type by carter, roger ebook. 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. The chapter introduces certain derivations d i of a kac moody algebra g, and then defines an extended algebra g e, obtained by adjoining such derivations to g.
A classification of lnvolutive automorphisms of an affine. Characters of representations of affine kacmoody lie algebras at the critical level, arxiv. Kac, basic representations of affine lie algebras and dual resonance models invent. David hernandez, an introduction to affine kacmoody algebras. We shall find it convenient to express the structure of an untwisted or twisted affine lie algebra in terms of formal variables. Kacmoody lie algebras, spectral sequences, and the witt. Includes material on lie algebras and on root systems that can be read independently. An indefinite matrix s gives rise to a kacmoody algebra of indefinite type. The spherical hecke algebra for affine kacmoody groups i. Characters of representations of affine kacmoody lie. 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. Lie algebras have many varied applications, both in mathematics and mathematical physics.
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