Moody, generalizing the finitedimensional semisimple lie alge bras which we refer to as the finite case. Groups and group functors attached to kacmoody data. The cohomology rings of kacmoody groups,their flag manifolds and classifying spacesj. S kumar kacmoody lie algebras 9 were introduced in the mid1960s independently by v.
Algebras of twisted chiral differential operators and. Unc mathematics department professor shrawan kumar. Affine kacmoody lie algebras and distinguished subalgebras. Thorbergsson 2003, these manifolds are the only nonhermitian symmetric.
The idea is to study the structure of each homogeneous space ln as a module over the affine kac moody lie algebra lo. Kacmoody groups, their flag varieties and representation. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way. Jun 25, 2007 in, we studied the flag variety of kac moody lie algebras, as a typical case of an infinitedimensional manifold. If time permit we shall also see some connections with theta functions. The notion of kacmoody lie algebra is a generalization of that of semisimple lie algebra to infinite dimension of the underlying vector space definition examples. Affine kacmoody algebras and semiinfinite flag manifolds. In this case, ga is just the simply connected complex semisimple lie group with cartan matrix a. Introduction kac moody algebras basic theory representation theory of kac moody algebras lie algebra homology and cohomology an introduction to indvarieties and progroups tits systems basic theory kac moody groups basic theory generalized flag varieties of kac moody groups demazure and weylkac character formulas. Intergrable representations of kacmoody lie algebras.
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 emphasis is on the study of the kacmoody groups 9 and their flag varieties xy, including their detailed construction, and their applications to the representation theory of g. Cartan matrices and their associated kacmoody lie algebras, kacmoody groups are divided into three types. Brion, lectures on the geometry of flag varieties, pdf. This lie algebra vdv contains the kac moody algebra a as a subalgebra but is always far larger than a. There is also a section that constructs a onedimensional extension of the loop group by loop rotation, which is a fairly transparent definition. An introduction to affine kacmoody lie algebras and groups 481 1. Since the terminology is not completely uniform even in the case of lie algebras, we start with brief discussion of this case. The soliton equations associated with the affine kacmoody. Thence we can build a real formofthewittalgebraas x. Proceedings of the jami inaugural conference, johns hopkins univ.
Ibe a symmetrizable generalized cartan matrix, g an associated kacmoody lie algebra, and t its cartan subalgebra. Kacmoody lie algebras where simultaneously introduced by kac and moody in the 1960s. This is applied to 1 finding the character of gintegrable irreducible highest weight modules over the affine lie algebra at the critical level, and 2 computing a certain elliptic genus of the flag manifold. Algebraic analysis, geometry and number theory, johns. By that study, it is revealed that the most natural. Introduction a riemannian manifold m,g is called homogeneous if it admits a transitive connected lie group g. We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Kashiwara, the flag manifold of kac moody lie algebra, algebraic analysis, geometry and number theory baltimore, md, 1988, johns hopkins univ. Kac and peterson 1983 infinite flag varieties and conjugacy theorems. Ams representation theory of the american mathematical society.
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. We consider the problem of diagonalization of the hamiltonians of the gaudin model, which is a quantum chain model associated to a simple lie algebra. 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. If youre reading this essay far in the future because youre interested in the topic i. The main tool is a result that interprets the drinfeldsokolov. Nonrepresentability of cohomology classes by biinvariant forms gauge and kac moody groups commu. It is shown that wakimoto modules introduced in 18, which are important in conformal field theory, correspond to certain sheaves on a semiinfinite flag manifold with support on its schhubert cells.
In, we studied the flag variety of kacmoody lie algebras, as a typical case of an infinitedimensional manifold. For the lie algebras of types a,b,c similar results were obtained by other methods by i. 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. Character formula for the intergrable highest weight modules. Then k acts transitively on any conjugacy class of parabolic subgroups, and hence the generalized flag variety gp is a compact homogeneous riemannian manifold kk. In the general case we get a lie algebra of vector. 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. The three preceding examples all give the same lie algebra structure. We study representations of affine kacmoody algebras from a geometric point of view.
Kashiwarathe flag manifold of kacmoody lie algebra. Grassmannians, schubert varieties, flag manifolds see also 32m10, 51m35 17b55. The indices a,b,c of the kacmoody extension are the lie algebra g indices ranging from 1,2,3,dim g where g is the corresponding lie algebra associated with the group g. Special issue on kacmoody algebras and applications. The nil hecke ring and cohomology of gp for a kac moody group g with b. Kacmoody lie algebras 9 were introduced in the mid1960s independently by v. Pdf on the poincar\e series of kacmoody lie algebras. They naturally generalise finite dimensional semisimple lie algebras. The generators of the sun kacmoody extension of the areapreserving di. Pdf a vertex algebra attached to the flag manifold and. This a ne lie algebra and its cousin, the kacmoody vertex algebra, are foundational objects in representation theory and conformal eld theory.
Vertex algebras, kacmoody algebras, and the monster. The analogous result for symmetrizable kacmoody lie algebras is the celebrated theorem of peterson and kac pk see also kmr and mp for detailed proofs. To reduce vdv to a smaller subalgebra, we will use the virasoro algebra. The most general kacmoody algebra of standard affine type is a semidirect. Each flag manifold carries a unique algebra of chiral differential operators. Uniqueness and presentation of kacmoody groups over fields. Intersection cohomologies of schubert varieties, operator algebras, unitary representations, enveloping algebras. Shrawan kumar, kacmoody groups, their flag varieties and representation theory, birkhauser 2002. This generalisation, appart from its own interest, has shown many applications in the finite dimensional setting. The flag manifold of kacmoody lie algebra, ps 21kb pdf 38kb. A homology vanishing theorem for kac moody algebras with coefficients in the category o. Finally, inchapter 3, i explain how the 2kacmoody algebra categori. Oct 20, 2017 this lie algebra comes with a cohomological grading, and for a given quiver q, there is a quiver with potential for which the degree zero part of the bps lie algebra is the positive half of the.
Kazhdanlusztig conjecture for a symmetrizable kacmoody lie. In case g is a kacmoody algebra, we have two versions of ag varieties xand x, that we call the thin ag variety and thick ag manifold, respectively see e. Moody, and subsequently many results of the theory of finitedimensional semisimple lie algebras have been carried over to kacmoody algebras. Abstract references similar articles additional information. These are the notes for a part iii course given in the university of cambridge in autumn 1998.
These algebras form a generalization of finitedimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such as its root system, irreducible representations, and connection to flag manifolds have natural analogues in the kacmoody setting. Lie groups, lie algebras, and generalized flag manifolds this is an informal introduction to lie groups. Parabolic kazhdanlusztig polynomials and schubert varieties. Proceedings of the jami inaugural conference, supplement to am.
Kacmoody groups, their flag varieties and representation theory. In 10wegive a geometric construction ofrepresentations ofthe kacmoody algebra. The pbw filtration introduction let g be a kacmoody lie. Pdf frobenius splitting of thick flag manifolds of kac. A lie algebra g is solvable if there exists a sequence of lie subalgebras 0. We study representations of affine kac moody algebras from a geometric point of view. The cohomology rings of kacmoody groups,their flag. Kacs book kac90l this selfcontained work treats the algebrogeometric and the topological aspects of kacmoody. This lie algebra vdv contains the kacmoody algebra a as a subalgebra but is always far larger than a. Kacmoody groups, their flag varieties, and representation theory. If the local action is by r, it gives rise to a vector. Wassermann, lecture notes on kacmoody and virasoro algebras. Grothendieck is again the most appropriate algebraic tool to deal with an infinitedimensional manifold. Kazhdanlusztig conjecture for a symmetrizable kacmoody.
Generalized kacmoody algebras are also sometimes called gkm algebras, borcherdskacmoody algebras, bkm algebras, or borcherds algebras. Lie groups richard borcherds, mark haiman, nicolai reshetikhin, vera serganova, and theo johnsonfreyd october 5, 2016. Opers on the projective line, flag manifolds and bethe. One can develop the general structure theory of complex semisimple lie. Kacmoody lie algebras 9 were introduced in the mid1960s independently by. The general results on relative lie algebra cohomology in 12d uniformly explain both garlands empirical equality and botts analogous classical strange equality, which kostant 11c had previously ex plained a different way. Gauge theories of kacmoody extensions w algebras as e. Kazhdanlusztig conjecture for a symnietrizable kacmoody lie. For the construction ofthe action ofits upper triangular part, we basically follow lusztigs idea l2, l3, but our approach is different from his in two points. The flag manifold of kacmoody lie algebra, algebraic analysis, geometry, and number theory. Then the lie bracket of two typical elements in gb of the type. Let h be the algebra ofpolynomialfunctions onthe cartan subalgebra h over. Let g be a semisimple lie group with maximal compact subgroup k. Kac moody lie algebras as bps lie algebras youtube.
Algebraic analysis, geometry and number theory, johns hopkins press, baltimore 1990 p. See 10, for instance, for thewx original theorem for a finitedimensional semisimple lie algebra, and 7wx and 8 for a kacwx moody algebra. Algebras of twisted chiral differential operators and affine. 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. Here is the most remarkable application of loop algebras.
Frobenius splitting of thick ag manifolds of kacmoody. Strictly speaking, g is the quotient of a certain kacmoody lie algebra by a onedimensional. Abstract the present paper studies the connection between the category of modules over the affine kac moody lie algebra at the critical level, and the category of dmodules on the affine flag scheme \gti\, where \i\ is the iwahori subgroup. 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 lie algebras in terms of parabolic kazhdanlusztig poly. The bethe ansatz equations can be written for an arbitrary kacmoody algebra, and we prove an analogue of the last result in this more general setting. Let gbe a semisimple lie group, hbe a cartan subgroup, wbe its corresponding weyl group with generators al,a2. Kashiwara, the flag manifold of kacmoody lie algebra, algebraic analysis, geometry and number theory baltimore, md, 1988, johns hopkins univ. Probably it is better to assume that the action is linear.
Schubert presentation of the cohomology ring of flag. Dmodules on the affine flag variety and representations of. Kacmoody algebras and the lie algebra of the line 8 2. Oct 02, 2012 in this paper, we discuss the poincar\e series of kacmoody lie algebras, especially for indefinite type. A systematic study of kacmoody algebras was started independently by v. We have chosen a different way to deal with these problems. By that study, it is revealed that the most natural language of scheme introduced by a. Firstly, we compute the poincar\e series of certain indefinite kacmoody lie. A detailed treatment of the lie algebra aspect of the theory can be found in v. Representations of quivers and preprojective algebras 9 3.
Pdf we explain that the pl\\ucker relations provide the defining equations of the thick flag manifold associated to a kacmoody algebra. Relaxed highest weight modules from dmodules on the kashiwara. The finitedimensional complex semisimple lie algebras. Affine kacmoody algebras a natural problem is to generalize the theory of. These algebras form a generalization of finitedimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. Kazhdanlusztig conjecture for symmetrizable kacmoody lie algebra ii. Kacmoody groups, their flag varieties, and representation. Kacmoody lie algebras 9 were introduced in the mid1960s independently. First, we have extra vector space wk for each vertex k.
Kazhdanlusztig conjecture for kacmoody lie algebras. The higher kacmoody analogs of the exceptional semisimple lie algebras e7, e7, e8 are. We obtain a corresponding dual semicanonical basis of cnw. Mar 15, 2002 we shall give a description of the intersection cohomology groups of the schubert varieties in partial flag manifolds over symmetrizable kacmoody lie algebras in terms of parabolic kazhdanlusztig polynomials introduced by deodhar. The ring of invariants for smooth completions of kacmoody lie algebras. Kacmoody lie algebras, spectral sequences, and the witt.