The indices a,b,c of the kacmoody extension are the lie algebra g indices ranging from. These lectures are intented to be a natural following of lectures on classical reprensentation theory. Dmodules on the affine grassmannian and representations of affine kacmoody algebras frenkel, edward and gaitsgory, dennis, duke mathematical journal, 2004. I wonder if any one has calculated the quantum dimensions of those kacmoody primary fields. Kacmoody algebras in 5 dimensional kaluzaklein theory. The pdf version contains the table of contents as bookmarks, which allows easy navigation. The theory has undergone tremendous developments in various directions and connections with diverse. Local geometric langlands correspondence and affine kacmoody. Oct 12, 2018 a considerable volume of works on the topic, the structure of general kacmoody algebras remains, to this day, largely mys terious. 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. We propose a conjectural description of these categories as categories of representations of the corresponding affine kac. Vertex algebras, kacmoody algebras, and the monster. For applications of the kacmoody algebra to physics see for instance dolan, goddard et al. Commutator relations and structure constants for rank 2 kacmoody.
On the homology of the kac moody groups and the cohomology of the 3connective covers of lie groups nishimura, osamu, journal of mathematics of kyoto university, 2002. We introduce a new class of infinitedimensional lie algebras, which arise as continuum colimits of borcherdskacmoody algebras. It serves as a reference book for physicists with commentary notes and reprints. Kacmoody algebras and lie algebras of regular vector fields on tori. Unlimited viewing of the articlechapter pdf and any associated supplements and figures. Kacmoody groups and cluster algebras christof geiss, bernard leclerc, and jan schroer abstract. By a local geometric langlands correspondence for a complex reductive group g we understand a construction which assigns to a local system on the punctured disc for the langlands dual group of g, a category equipped with an action of the formal loop group gt. Higher kacmoody algebras and symmetries of holomorphic field.
There is also a generalized kac moody algebra associated to any even lorentzian lattice of dimension at most 26 or any lorentzian lattice. These lectures were held at the graduate course on conformal field. The cft of the sun level k kac moody current algebra has many kac moody primary fields. We analyze the extent to which a quantum universal enveloping algebra of a kacmoody algebra g is defined by multidegrees of its defining relations. We give a construction of the tits functor for symmetrizable kac moody groups g using integrable highest weight modules for the corresponding kac moody algebra and a zform of the universal enveloping algebra. We construct a 2category associated with a kac moody algebra and we study its 2representations. 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 monstrous generalized kacmoody algebra g z has as root lattice the even 26 dimensional unimodular lorentzian lattice. Affine kacmoody algebras a natural problem is to generalize the theory of. Kacmoody lie algebras, spectral sequences, and the witt.
For applications of the kac moody algebra to physics see for instance dolan, goddard et al. Communicated by walter feit, december, 1985 abstract it is known that the adjoint representation of any kacmoody. The monstrous generalized kac moody algebra g z has as root lattice the even 26 dimensional unimodular lorentzian lattice. A lie algebra g is a generalized kacmoody algebra if it satis. We describe the structure of a hyperbolic kacmoody algebra of rank 2 treated as a module over a root subalgebra which is isomorphic to a simple 3dimensional lie algebra and is. Uniqueness and presentation of kacmoody groups over fields core. Central extensions of generalized kacmoody algebras. Representation theory of superconformal algebras and the kacroanwakimoto conjecture arakawa, tomoyuki, duke mathematical journal, 01 december 05. I wonder if any one has calculated the quantum dimensions of those kac moody primary fields. Quantizations of kacmoody algebras quantizations of kacmoody algebras kharchenko, v. The corresponding lie algebras ga the kacmoody algebras were introduced in. The gkm algebra g associated to this is defined to be the lie algebra generated by h and elements ei and fi for i in. I am trying to make sense to the issue of how does the kac moody algebra encode the symmetries of the nontruncated theory lets contextualize a little bit.
Generalized kac moody algebras are also sometimes called gkm algebras, borcherds kac moody algebras, bkm algebras, or borcherds algebras. In this article we study the cohomology of k as a hopf algebra. We introduce an affinization of the quantum kac moody algebra associated to a symmetric generalized cartan matrix. In particular, if f has positive characteristic, we show that h. Semicanonical bases arising from enveloping algebras. A homology vanishing theorem for kac moody algebras with coefficients in the category o.
Some remarks on triangulated categories and kacmoody. In the absence of auxiliary fields in the wesszumino model, supersymmetry transformations generate an infinitedimensional lie algebra, which is shown to be a grassmannian extension of this kacmoody algebra. A lie algebra with lie bracket 23 is called an untwisted affine kacmoody algebra. 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. 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. 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. Citeseerx infinite dimensional chevalley groups and kac. In the finite case, 9 is nothing but a semisimple y simplyconnected algebraic group and x is the flag variety 9 py for a parabolic subgroup p y c g.
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. We construct a 2category associated with a kacmoody algebra and we study its 2representations. 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. Borcherds, trinity college, cambridge cb2 1tq, england. Kac moody groups and cluster algebras christof geiss, bernard leclerc, and jan schroer abstract. In this paper, we classify the module structures on u h. Pdf quantum kac moody algebras and vertex representations. This gives a construction of kac moody groups g as infinite dimensional analogs of finite dimensional chevalley groups. For the time being we will focus in the case that c. Moody, generalizing the finitedimensional semisimple lie alge bras which we refer to as the finite case. These are the notes for a part iii course given in the university of cambridge in autumn 1998. Quantum kac moody algebras and vertex representations. Download pdf introduction to kac moody algebra free. The idea is to study the structure of each homogeneous space ln as a module over the affine kacmoody lie algebra lo.
The idea is to study the structure of each homogeneous space ln as a module over the affine kac moody lie algebra lo. Representation theory of superconformal algebras and the kac roanwakimoto conjecture arakawa, tomoyuki, duke mathematical journal, 01 december 05. Xiao, triangulated categories and kacmoody algebras, invent. Pdf on jan 1, 1994, jacqui ramagge and others published an introduction to kacmoody groups find, read and cite. Gauge theories of kacmoody extensions w algebras as e. The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations.
This gives a construction of kacmoody groups g as infinite dimensional analogs of finite dimensional chevalley groups. Usually, the definitions presented involve the use of. Kacmoody lie algebras where simultaneously introduced by kac and moody in the 1960s. New representations of affine kacmoody algebras sciencedirect. The cartan matrix aalso determines the lie algebra associated to gvia serres presentation. To this end, we consider a class of character hopf algebras defined by the same number of defining relations of the same degrees as. Reference on highest weight module of kac moody algebra. Pdf on the structure of kacmoody algebras researchgate. The nil hecke ring and cohomology of gp for a kac moody group g with b. 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.
A characterization of generalized kacmoody algebras. Solution of the initial value problem for the sinegordon equation using a kacmoody algebra. Algebra 445 2016 232243 download pdf presentation of affine kacmoody groups over rings algebra and number theory 10 2016 533556 download pdf geometric generators for braidlike groups with tathagata basak geometry and topology 20 2016 747778. These algebras form a generalization of finitedimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. Quantizations of kacmoody algebras, journal of pure and. Moody, and subsequently many results of the theory of finitedimensional semisimple lie algebras have been carried over to kacmoody algebras. We describe the structure of a hyperbolic kac moody algebra of rank 2 treated as a module over a root subalgebra which is isomorphic to a simple 3dimensional lie algebra and is associated with an imaginary root. We nish by discussing the structure and representation theory of reductive algebraic groups. Generalized kacmoody algebras are also sometimes called gkm algebras, borcherdskacmoody algebras, bkm algebras, or borcherds algebras. Kac moody groups, their flag varieties and representation.
Pdf an introduction to kacmoody groups researchgate. Kacmoody algebras are a generalization of the finitedimensional semisimple. The problem of classification of simple finite dimensional lie algebras over the field. 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. There is a characterization of generalized kacmoody algebras given in 3 as follows. Nonrepresentability of cohomology classes by biinvariant forms gauge and kac moody groups commu. There is also a generalized kacmoody algebra associated to any even lorentzian lattice of dimension at most 26 or any lorentzian lattice. Generalized kacmoody algebras connecting repositories. Click download or read online button to introduction to kac moody algebra book pdf for free now. 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. In mathematics, a generalized kacmoody algebra is a lie algebra that is similar to a kac moody algebra, except that it is allowed to have imaginary simple roots. Download pdf presentation of hyperbolic kacmoody groups over rings with lisa carbone, j. We consider the problem of representing the kac moody algebra \mathfrakgn specified by an r\times r. Kacmoody groups, their flag varieties and representation.
View the article pdf and any associated supplements and figures for a period of 48 hours. A lie algebra with lie bracket 23 is called an untwisted affine kac moody algebra. We give a construction of the tits functor for symmetrizable kacmoody groups g using integrable highest weight modules for the corresponding kacmoody algebra and a zform of the universal enveloping algebra. The cft of the sun level k kacmoody current algebra has many kacmoody primary fields. Representations of kacmoody algebras and dual resonance models. Reference on highest weight module of kacmoody algebra. In particular, this theorem gives a global realization of symmetrizable kacmoody algebras. Bgg categories of a kacmoody lie superalgebra and a kacmoody lie algebra. We shall not need more that basic notions on linear algebra and lie algebras. Kac moody lie algebras 9 were introduced in the mid1960s independently by v. Introduction to kac moody algebra download introduction to kac moody algebra ebook pdf or read online books in pdf, epub, and mobi format. In this paper, we probe deeply into the structure of those algebras the e. A hyperbolic kacmoody algebra and the theory of siegel.
This generalizes earlier work with chuang for type a. Dmodules on the affine grassmannian and representations of affine kac moody algebras frenkel, edward and gaitsgory, dennis, duke mathematical journal, 2004. Local geometric langlands correspondence and affine kac. Lie algebras which preserve almost the full original structure. Aug 20, 2005 by a local geometric langlands correspondence for a complex reductive group g we understand a construction which assigns to a local system on the punctured disc for the langlands dual group of g, a category equipped with an action of the formal loop group gt.
Kacmoody group over c, let k denote the unitary form with maximal torus t having normalizer nt. A hyperbolic kacmoody algebra and the theory of siegel modular forms of genus 2. Hyperbolic kacmoody algebras of rank 2 springerlink. A systematic study of kacmoody algebras was started independently by v. In particular, on any complex manifold there is a factorization algebra of currents associated to any lie algebra. 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 combinatorics of category o over symmetrizable kacmoody. Irreducible characters of kacmoody lie superalgebras. In this paper, we introduce a new class of imaginary roots, called purely imaginary roots and give a necessary and sufficient condition for an imaginary root to be purely imaginary,thereby giving a complete classification of those kac moody algebras with the purely imaginary property, namely, all their imaginary roots are purely imaginary. The emphasis is on the study of the kac moody groups 9 and their flag varieties xy, including their detailed construction, and their applications to the representation theory of g.
The invariance algebra of the majorana action contains a kacmoody algebra which, on shell, reduces to an abelian algebra. The problem of attaching groups to such algebras has. Kacmoody lie algebras 9 were introduced in the mid1960s independently by v. Introduction this paper provides a detailed exposition of the theory of generalized kacmoody algebras, whose theory was originally developed by r. We propose a conjectural description of these categories as categories of representations of the corresponding affine kac moody. Home representations of kac moody algebras and dual resonance models. In this paper, we introduce a new class of imaginary roots, called purely imaginary roots and give a necessary and sufficient condition for an imaginary root to be purely imaginary,thereby giving a complete classification of those kacmoody algebras with the purely imaginary property, namely, all their imaginary roots are purely imaginary. Representations of kac moody algebras and dual resonance models. Introduction this paper provides a detailed exposition of the theory of generalized kac moody algebras, whose theory was originally developed by r. We completely determine the structure constants between real root vectors in rank 2 kacmoody algebras. In the 1960s, serre demonstrated that each semisimple lie algebra could be defined up to an isomorphism by a finite number of generators and commutation. 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. 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.
Sinegordonequationandrepresentationsof affine kac moody. Affine kacmoody algebras, chl strings and the classification of. Ok, so in the 5 dimensional kaluzaklein we start with pure 5 dimensional gravity. One exception is the case where all imaginary ro ots.
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