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