Algebraic analysis, geometry, and number theory baltimore, md, 1988, 161190, johns hopkins univ. An indefinite matrix s gives rise to a kacmoody algebra of indefinite type. Pdf irreducible representations of untwisted affine kacmoody. In mathematics, a kacmoody algebra is a lie algebra, usually infinite dimensional, that can be. The soliton equations associated with the affine kacmoody lie algebra g 2 1.
Pdf in this paper we construct a class of new irreducible modules over the. Kacmoody lie algebras, spectral sequences, and the witt. Pdf a characterization of affine kacmoody lie algebras. This program is related to topology, and prob ably physics,1 as well as combinatorics. A positive semidefinite matrix s gives rise to an infinitedimensional kacmoody algebra of affine type, or an affine lie algebra. Does anyone know a simple construction for affine kac moody groups. 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. Pdf the soliton equations associated with the affine kac. The following references discuss aspects of the kacmoody exceptional geometry of supergravity theories. Character formula for the intergrable highest weight modules. The book will be useful to lie algebraists, high energy physicists, statistical mechanics, and number theorists.
The finite type indecomposable kac moody algebras are precisely the finitedimensional simple lie algebras. Determining a consistent system of signs of structure constants in lie algebras and kacmoody algebras is a persistent problem in lie theory, computational algebra, computational number theory and their applications. Among kacmoody lie algebras, affine lie algebras are an important infinite dimensional class. Affine lie algebras are particular examples of lie algebras defined by cartan matrices, or kacmoody lie algebras. An indefinite matrix s gives rise to a kac moody algebra of indefinite type. In this paper, an algorithm for computing the principal character for a ne lie algebras is discussed and presented. 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. Kacmoody lie algebras graded by kacmoody root systems. David hernandez, an introduction to affine kacmoody algebras. Volume one contains a description of kac moody lie algebras, and especially the affine algebras and their representations.
On an analog of the onsager algebra of type d1n 43 54. In these lectures the untwisted affine lie algebras will be constructed. Lie algebras which preserve almost the full original structure. We extend, to general kacmoody lie algebras, the notion of cadmissible pair as introduced by h. Antony wassermann, kacmoody and virasoro algebras, course notes 2011. A classification of lnvolutive automorphisms of an affine kac. We shall not need more that basic notions on linear algebra and lie algebras.
This paper deals with the problem of conjugacy of cartan subalgebras for a. Most of the important applications of kac moody theory so far center around this. We give a new characterization of the affine kac moody algebras in terms of extended affine lie algebras. The aim of this chapter is to explicitly realize the kac moody algebras of affine type also called the affine kac moody algebras and the associated groups. The three preceding examples all give the same lie algebra structure. These lectures are intented to be a natural following of lectures on classical reprensentation theory. David hernandez, an introduction to affine kacmoody algebras 2006 iain gordon, infinitedimensional lie algebras 20089 pdf. A kacmoody algebra also kacmoody lie algebra is defined as follows. We give a new characterization of the affine kacmoody algebras in terms of extended affine lie algebras. The idea is to study the structure of each homogeneous space ln as a module over the affine kac moody lie algebra lo. Lie algebras, fuchsian differential equations and cft correlation functions 115 126.
Results include level 2 and 3 character formulas in a2 2n 1 and the sole existence of the rogers. We also present new realizations of the twisted affine kac moody algebras. A characterization of affine kacmoody lie algebras. A description of automorphism of kacmoody algebras is given by kac and wang 11 and cornwell et al 21, 22. There is a book by kumar kac moody groups, their flag varieties, and representation theory that does the construction for the general kac moody case, but i find the presentation dense. The principal characters discovered using this program are given andor proven. Introduction the hyperbolic kac moody lie algebras have been considered as the next natural objects of study after the affine case.
If time permit we shall also see some connections with theta functions. Thence we can build a real formofthewittalgebraas x. Real forms of affine kac moody lie algebras are either almost split or almost compact. On the tensor semigroup of affine kacmoody lie algebras. A class of kacmoody algebras called affine lie algebras is of particular. A positive semidefinite matrix s gives rise to an infinitedimensional kac moody algebra of affine type, or an affine lie algebra. Kac, basic representations of affine lie algebras and dual resonance models invent. Kacmoody chevalley groups and lie algebras with builtin. Kacmoody algebras are a generalization of the finitedimensional semisimple. 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. An introduction to affine kac moody lie algebras and groups. In this paper, we are interested in the decomposition of the tensor product of two representations of a symmetrizable kacmoody lie. Ja977125 lieadmissible algebras and kacmoody algebras kyeonghoon jeong, seokjin kang, and hyeonmi lee department of mathematics, seoul national uniersity, seoul, 151742, korea.
Intergrable representations of kacmoody lie algebras. Conformalfieldtheoretic analogues of codes and lattices 1 142. Affine kacmoody algebras a natural problem is to generalize the theory of. The idea is to study the structure of each homogeneous space l as a module over the affine kacmoody lie algebra lq. 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. On the hyperbolic kacmoody lie algebra ha1 1 source. These are the notes for a part iii course given in the university of cambridge in autumn 1998. Real forms of affine kacmoody lie algebras are either almost split or almost compact. In particular, when lq is an affine kacmoody lie algebra and v is the basic representation of l0, we obtain a series of lorentzian kacmoody lie algebras e. Explicit construction of finite dimensional lie algebras. Oof the langlands dual kac moody algebra in terms of bradenmacpherson sheaves on the moment graph. Since the diagonal entries of c and s are positive, s cannot be negative definite or negative semidefinite. 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.
Subalgebras of hyperbolic kac moody algebras 97 108. We also present new realizations of the twisted affine kacmoody algebras. It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. Victor kac, infinite dimensional lie algebras, lecture notes include. Lie algebras associated with generalized cartan matrices pdf. Pdf an introduction to kacmoody groups researchgate. Fundamental representations of the lie algebra of infinite matrices.
The collection of indecomposable kacmoody algebras is divided into three mutually exclusive types. System upgrade on feb 12th during this period, ecommerce and registration of new users may not be available for up to 12 hours. Affine lie algebras, weight multiplicities, and branching. Invariants of real forms of affine kacmoody lie algebras. As far as applications this is the most important class. Affine kacmoody algebras, integrable systems and their deformations 3 of x, equipped with a coordinate t, the space of sl nopers on u may be identi.
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