New fusion products for the affine kacmoody algebra sl. Vertex algebras, kacmoody algebras, and the monster. 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. Di xu ucsc virasoro and kacmoody algebra 20150611 14 24. Some definitions we will need to introduce kacmoody algebras. The notion of kacmoody lie algebra is a generalization of that of semisimple. Secondly, we are also restricting ourselves to symmetrizable kacmoody algebras. A proof of the character formula for a standard module is given, generalizing the argument of garland and lepowsky for the kacmoody case.
Generalized kacmoody algebras, automorphic forms and. A matrix ais a generalized cartan matrix gcm if it satis es. It is known that the adjoint representation of any kacmoody algebra a can be identified with a subquotient of a certain fock space representation constructed from the root lattice of a. 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 invariance algebra of the majorana action contains a kacmoody algebra which, on shell, reduces to an abelian algebra.
An exposition of generalized kacmoody algebras elizabeth jurisich abstract. Pdf on the structure of kacmoody algebras researchgate. These algebras form a generalization of finitedimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. In the basis of eigenstates of the two operators l0 and l0 are denote eigenvalues by h. Kac moody algebras and the lie algebra of the line 8 2. A special class of indefinite type of kacmoody algebra, called an extendedhyperbolic kacmoody algebra and the. The purpose of this essay is to define the 2kacmoody algebra 9uqpgq and prove that this definition categorifies the integral idempotent. A hyperbolic kacmoody algebra and the theory of siegel. Communicated by walter feit, december, 1985 abstract it is known that the adjoint representation of any kacmoody. A lie algebra g is called a generalized kacmoody algebra if g is the semidirect product. Characterizations of a generalized kacmoody algebra.
Vertex algebras, kacmoody algebras, and the monster pnas. Representations of kacmoody algebras and combinatorics. Kacmoody lie algebras where simultaneously introduced by kac and moody in the 1960s. This subsumes the case of affine lie algebras in which case the corresponding kacmoody group is a centrally extended loop group see there for more and often the words kacmoody group are used synonomously with centrally extended loop group. This product together with a infinite number of other products is constructed using a. Journal of algebra 115, 501512 1988 generalized kacmoody algebras richard borcherds department qf pure mathematics and mathematical sratistics. The idea is to study the structure of each homogeneous space ln as a module over the affine kacmoody lie algebra lo. Affine kacmoody algebras a natural problem is to generalize the theory of.
The simplest indefinite kacmoody algebras are the rank 2 kacmoody algebras a a. We shall not need more that basic notions on linear algebra and lie algebras. Some remarks on triangulated categories and kacmoody. Generalized kacmoody algebras are also sometimes called gkm algebras, borcherdskacmoody algebras, bkm algebras, or borcherds algebras. A ne kacmoody algebras and the wesszuminowitten model. Introduction over the years the groundstate ensembles of certain twodimensional classical spin models have been found to exhibit critical behavior. As with the finite dimensional lie algebras, these have been well studied over the years and have.
The root lattice l of g is the even unimodular lorentzian lattice ii1. Both lie algebras are fundamental in mathematics and. 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. Kacmoody groups and cluster algebras christof geiss, bernard leclerc, and jan schroer abstract. When the billiard is associated with a kacmoody algebra, the criterium for. These are the notes for a part iii course given in the university of cambridge in autumn 1998. For a symmetrizable kacmoody algebra, we can define a normalized invariant form that performs the same role as the killing form in the finite dimensional case. 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. Automorphic correction of the hyperbolic kacmoody algebra. A kacmoody algebra is symmetrizable if there exists a non degenerate.
Lie algebras which preserve almost the full original structure. Thence we can build a real formofthewittalgebraas x. A group corresponding to a kacmoody algebra is a kacmoody group. These lectures are intented to be a natural following of lectures on classical reprensentation theory. The main reason for this is that they can be realized as the universal onedimensional central extension of the lie algebra of laurent maps from to a semisimple lie algebra. Theweight space ofthe representation space will be given as a.
Iv, we obtain automorphic correction of the hyperbolic kacmoody algebra e 10. Kacmoody algebras and the lie algebra of the line 8 2. 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. 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. The indices a,b,c of the kacmoody extension are the lie algebra g indices ranging from.
Kac 2, introduced the concepts of strictly imaginary roots for kacmoody algebras. Critical ground state, kac moody algebra, coulomb gas 1. Integrable modules for each i 1n, let s i be the lie subalgebra with vector space basis fe i. The idea is to study the structure of each homogeneous space l as a module over the affine kacmoody lie algebra lq. Affine kacmoody algebras let g be a simple lie algebra over c. Motivated by his results, we give a geometric construction of irreducible highestweight integrable representations of the kacmoody algebra associated to the graph theorem 10. 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. Borcherds, trinity college, cambridge cb2 1tq, england. Finally, it is worthy of relating the work of the present paper to stability conditions on triangulated categories defined by bridgeland in.
Then s i is isomorphic to sl 2, and s i acts on ga via the adjoint action. In section 1 we give a brief reminder of the definition of a kacmoody algebra gi associated to a graph i. The first such groundstate ensemble was encountered in the antiferromagnetic ising model on the triangular lattice 1,2. The continuum kacmoody algebras of the line a nd the circle. In the 1970s, high energy physicists pursued lie algebra theory as a valuable tool to characterize all the gauge interactions. Note that questions of realizing the entire kacmoody lie algebra g, u v g and u g through the associative hall type algebra over triangulated categories still remain open. One of the key objects from nitedimensional theory is the cartan matrix which encodes everything necessary to recover the lie algebra from the chevalleyserre 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. What is an intuitive explanation of kacmoody algebras. It serves as a reference book for physicists with commentary notes and reprints. Let m be the unique maximal ideal which intersects h trivially. Kacmoody and virasoro algebras, course notes 2011 pdf. This is the kac moody algebra with cartan matrix a. By relaxing its axioms we will see in detail how the kac moody algebra is constructed and characterize its dimensionality through familiar notions. In this paper, we probe deeply into the structure of those algebras a.
A kacmoody algebra is a complex lie algebra generated by. The automorphic correction is provided by a borcherds lift on o10, 2 of the. Affine algebras are a subclass of the kacmoody algebras. New fusion products for the a ne kacmoody algebra slb 2 at level 12. Casperson 1, obtained the complete classification of kacmoody algebras possessing strictly imaginary property. 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. If youre reading this essay far in the future because youre interested in the topic i. Vertex algebra constructions of generalized kacmoody.
Finally, inchapter 3, i explain how the 2kacmoody algebra categori. These lecture notes are a brief introduction to wesszuminowitten models, and their current algebras, the a ne kacmoody algebras. Generalized mckay quivers, root system and kacmoody. I define a product on the whole of the fock space that restricts to the lie algebra product on this subquotient. The monster algebra g is a generalized kacmoody algebra describing the physical states of a bosonic string moving on a z2ztwisted orbifold. In mathematical terms, this means that both the ln and the. The three preceding examples all give the same lie algebra structure. Rank 2 symmetric hyperbolic kacmoody algebras nagoya. Kacmoody algebras are a generalization of the finitedimensional semisimple. The global conformal algebra is also useful for characterizing properties of physical states. For the time being we will focus in the case that c.
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. Now let a be an arbitrary symmetrizable irreducible cartan matrix. Kacmoody lie algebras, spectral sequences, and the witt. The kacmoody algebra gi is the lie algebra with generators ei,fi,hi.
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