The flag manifold of kac-moody lie algebra pdf

The three preceding examples all give the same lie algebra structure. In, we studied the flag variety of kacmoody lie algebras, as a typical case of an infinitedimensional manifold. Let gbe a semisimple lie group, hbe a cartan subgroup, wbe its corresponding weyl group with generators al,a2. 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. Let g be a semisimple lie group with maximal compact subgroup k. 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.

Lie groups, lie algebras, and generalized flag manifolds this is an informal introduction to lie groups. Proceedings of the jami inaugural conference, johns hopkins univ. Grassmannians, schubert varieties, flag manifolds see also 32m10, 51m35 17b55. 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. Grothendieck is again the most appropriate algebraic tool to deal with an infinitedimensional manifold. 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. The cohomology rings of kacmoody groups,their flag. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way. Kazhdanlusztig conjecture for a symnietrizable kacmoody lie. See 10, for instance, for thewx original theorem for a finitedimensional semisimple lie algebra, and 7wx and 8 for a kacwx moody algebra. The flag manifold of kacmoody lie algebra, algebraic analysis, geometry, and number theory. They naturally generalise finite dimensional semisimple lie algebras. In 10wegive a geometric construction ofrepresentations ofthe kacmoody algebra.

Representations of quivers and preprojective algebras 9 3. Kacmoody groups, their flag varieties and representation theory. Moody, and subsequently many results of the theory of finitedimensional semisimple lie algebras have been carried over to kacmoody algebras. 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. That is, they again give rise to a lie algebra, but there appears an additional generator c. 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. Affine kacmoody algebras a natural problem is to generalize the theory of. A homology vanishing theorem for kac moody algebras with coefficients in the category o. By that study, it is revealed that the most natural.

It is intended to make the topics of the school accessible to the participants who never took a course in lie groups and lie algebras but do know some pointset topology and some basic notions about di. Kashiwarathe flag manifold of kacmoody lie algebra. The flag manifold of kacmoody lie algebra, ps 21kb pdf 38kb. The soliton equations associated with the affine kacmoody. Kacmoody lie algebras in terms of parabolic kazhdanlusztig poly. Kacmoody lie algebras where simultaneously introduced by kac and moody in the 1960s. Pdf on the poincar\e series of kacmoody lie algebras. A detailed treatment of the lie algebra aspect of the theory can be found in v. Kacmoody lie algebras 9 were introduced in the mid1960s independently by v. Strictly speaking, g is the quotient of a certain kacmoody lie algebra by a onedimensional. Thence we can build a real formofthewittalgebraas x. 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. There is also a section that constructs a onedimensional extension of the loop group by loop rotation, which is a fairly transparent definition. In this case, ga is just the simply connected complex semisimple lie group with cartan matrix a.

Firstly, we compute the poincar\e series of certain indefinite kacmoody lie. Ams representation theory of the american mathematical society. The pbw filtration introduction let g be a kacmoody lie. Kashiwara, the flag manifold of kac moody lie algebra, algebraic analysis, geometry and number theory baltimore, md, 1988, johns hopkins univ. Kacmoody lie algebras, spectral sequences, and the witt. 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. 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. Cartan matrices and their associated kacmoody lie algebras, kacmoody groups are divided into three types. Kazhdanlusztig conjecture for a symmetrizable kacmoody. Then the lie bracket of two typical elements in gb of the type.

If time permit we shall also see some connections with theta functions. Vertex algebras, kacmoody algebras, and the monster. 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. This lie algebra vdv contains the kacmoody algebra a as a subalgebra but is always far larger than a. Kazhdanlusztig conjecture for kacmoody lie algebras. To reduce vdv to a smaller subalgebra, we will use the virasoro algebra. 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.

This lie algebra vdv contains the kac moody algebra a as a subalgebra but is always far larger than a. Kazhdanlusztig conjecture for a symmetrizable kacmoody lie. Kacmoody groups, their flag varieties, and representation theory. The generators of the sun kacmoody extension of the areapreserving di. Kac moody lie algebras as bps lie algebras youtube. Pdf a vertex algebra attached to the flag manifold and. One can develop the general structure theory of complex semisimple lie. Groups and group functors attached to kacmoody data. Brion, lectures on the geometry of flag varieties, pdf. Dmodules on the affine flag variety and representations of. Gauge theories of kacmoody extensions w algebras as e. 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 the general case we get a lie algebra of vector. 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.

Opers on the projective line, flag manifolds and bethe. 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. Kacmoody groups, their flag varieties, and representation. This generalisation, appart from its own interest, has shown many applications in the finite dimensional setting.

S kumar kacmoody lie algebras 9 were introduced in the mid1960s independently by v. This a ne lie algebra and its cousin, the kacmoody vertex algebra, are foundational objects in representation theory and conformal eld theory. Let g be a kacmoody lie algebra of finite or affine type and let ug be the. The idea is to study the structure of each homogeneous space ln as a module over the affine kac moody lie algebra lo. Kacmoody lie algebras 9 were introduced in the mid1960s independently by. Since the terminology is not completely uniform even in the case of lie algebras, we start with brief discussion of this case. 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. Kacmoody groups, their flag varieties and representation. Frobenius splitting of thick ag manifolds of kacmoody. Lie groups richard borcherds, mark haiman, nicolai reshetikhin, vera serganova, and theo johnsonfreyd october 5, 2016. First, we have extra vector space wk for each vertex k. Jun 25, 2007 in, we studied the flag variety of kac moody lie algebras, as a typical case of an infinitedimensional manifold.

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. Oct 02, 2012 in this paper, we discuss the poincar\e series of kacmoody lie algebras, especially for indefinite type. Introduction a riemannian manifold m,g is called homogeneous if it admits a transitive connected lie group g. Algebras of twisted chiral differential operators and. Kazhdanlusztig conjecture for symmetrizable kacmoody lie algebra ii. Citeseerx document details isaac councill, lee giles, pradeep teregowda. We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. If the local action is by r, it gives rise to a vector. We study representations of affine kac moody algebras from a geometric point of view.

The nil hecke ring and cohomology of gp for a kac moody group g with b. The cohomology rings of kacmoody groups,their flag manifolds and classifying spacesj. The main tool is a result that interprets the drinfeldsokolov. Moody, generalizing the finitedimensional semisimple lie alge bras which we refer to as the finite case. Here is the most remarkable application of loop algebras. By that study, it is revealed that the most natural language of scheme introduced by a. An introduction to affine kacmoody lie algebras and groups 481 1. 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. These are the notes for a part iii course given in the university of cambridge in autumn 1998. Relaxed highest weight modules from dmodules on the kashiwara. Parabolic kazhdanlusztig polynomials and schubert varieties. Algebraic analysis, geometry and number theory, johns.

Kacmoody lie algebras 9 were introduced in the mid1960s independently. Affine kacmoody lie algebras and distinguished subalgebras. Algebras of twisted chiral differential operators and affine. Probably it is better to assume that the action is linear. A systematic study of kacmoody algebras was started independently by v. Kacs book kac90l this selfcontained work treats the algebrogeometric and the topological aspects of kacmoody. We study representations of affine kacmoody algebras from a geometric point of view. For the lie algebras of types a,b,c similar results were obtained by other methods by i. A lie algebra g is solvable if there exists a sequence of lie subalgebras 0.

Generalized kacmoody algebras are also sometimes called gkm algebras, borcherdskacmoody algebras, bkm algebras, or borcherds algebras. Intergrable representations of kacmoody lie algebras. Uniqueness and presentation of kacmoody groups over fields. 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. Pdf we explain that the pl\\ucker relations provide the defining equations of the thick flag manifold associated to a kacmoody algebra. Nonrepresentability of cohomology classes by biinvariant forms gauge and kac moody groups commu. Affine kacmoody algebras and semiinfinite flag manifolds. Shrawan kumar, kacmoody groups, their flag varieties and representation theory, birkhauser 2002. The most general kacmoody algebra of standard affine type is a semidirect.

Special issue on kacmoody algebras and applications. Unc mathematics department professor shrawan kumar. Character formula for the intergrable highest weight modules. Ibe a symmetrizable generalized cartan matrix, g an associated kacmoody lie algebra, and t its cartan subalgebra. These algebras form a generalization of finitedimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. Kacmoody algebras and the lie algebra of the line 8 2. Let h be the algebra ofpolynomialfunctions onthe cartan subalgebra h over.

Algebraic analysis, geometry and number theory, johns hopkins press, baltimore 1990 p. 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. Finally, inchapter 3, i explain how the 2kacmoody algebra categori. 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. The ring of invariants for smooth completions of kacmoody lie algebras. If youre reading this essay far in the future because youre interested in the topic i. 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.

Schubert presentation of the cohomology ring of flag. Thorbergsson 2003, these manifolds are the only nonhermitian symmetric. The finitedimensional complex semisimple lie algebras. Kac and peterson 1983 infinite flag varieties and conjugacy theorems. The present paper studies the connection between the category of modules over the affine kacmoody lie algebra at the critical level, and the category of dmodules on the affine flag scheme, where is the iwahori subgroup. 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. Intersection cohomologies of schubert varieties, operator algebras, unitary representations, enveloping algebras. 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. Pdf frobenius splitting of thick flag manifolds of kac. The higher kacmoody analogs of the exceptional semisimple lie algebras e7, e7, e8 are. Abstract references similar articles additional information. Proceedings of the jami inaugural conference, supplement to am.