A completion of the quantized enveloping algebra of a kacmoody algebra. In mathematics, a kacmoody algebra is a lie algebra, usually infinitedimensional, that can be. Some subalgebras of indefinite type kacmoody lie algebras article in journal of pure and applied algebra 2125. Gauge theories of kacmoody extensions w algebras as e. Chariannihilators of verma modules for kacmoody lie algebras. These algebras form a generalization of finitedimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. Combinatorics in affine flag varieties sydney mathematics and.
Integrable representations and the weyl group of a kacmoody algebra. This survey paper presents the discrete group viewpoint on kacmoody groups. Kacmoody lie algebras 9 were introduced in the mid1960s independently by v. In mathematics, a kacmoody algebra is a lie algebra, usually infinite dimensional, that can be. Kacmoody groups, their flag varieties and representation theory. Generalized kacmoody lie algebras, free lie algebras and. Kacmoody lie algebras in terms of parabolic kazhdanlusztig poly. The theory of lie groups and their representations is a vast subject bourbaki bou. Chariannihilators of verma modules for kacmoody lie. 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. From purely mathematical point of view, affine lie algebras are interesting because their representation theory, like.
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 simplest models for the sun cosets involve the graphs that represent the fu sion by the fundamental representation in the kac. This generalisation, appart from its own interest, has shown many applications in the finite dimensional setting. It is a sheaf of vertex algebras in the zarisky topology, i. A proof of the character formula for a standard module is given, generalizing the argument of garland and lepowsky for the kac moody case. A lie algebra l is simple if it has no nontrivial ideals and it is not abelian. A, where g is any kac moo dy algebra and a is a r ing of laurent polynomials in n commuting variables. Infinitedimensional lie algebras, theta functions and. Part iii lecture notes on operators on hilbert space michaelmas 1991, cambridge. To reduce vdv to a smaller subalgebra, we will use the virasoro algebra. We present a detailed exposition of the theory of generalized kacmoody algebras associated to symmetrizable matrices. Kacmoody lie algebras, spectral sequences, and the witt.
There is also a section that constructs a onedimensional extension of the loop group by loop rotation, which is a fairly transparent definition. In the finite case, 9 is nothing but a semisimple y simplyconnected algebraic group and x is the flag variety 9. Notations all algebras under consideration in this paper are kacmoody lie algebras. On important new feature in this setting is the appearence of imaginary roots i. Since the terminology is not completely uniform even in the case of lie algebras, we start with brief discussion of this case. Lie algebras associated with generalized cartan matrices pdf. Let v kg be the universal vertex algebra generated by elds xz, x 2g. The theory of lie groups and their representations is a vast subject bourbaki. Zeng journal of pure and applied algebra 212 2008 11051115 2. Absolutely indecomposable representations and kacmoody. Concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations, this is the third revision of an important monograph. In drawing our dynkin diagrams, we use the conventions given in 4, sections 4. Kacmoody algebras, which, along with the virasoro algebras, are.
Vertex algebras and algebraic curves, second edition pdf. See 10, for instance, for thewx original theorem for a finitedimensional semisimple lie. Also, i have done the classifications of classical lie algebras. Consider the short exact sequence of lie algebra homomorphisms 0. 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. A proof of the character formula for a standard module is given, generalizing the argument of garland and lepowsky for the kacmoody case. Each node should correspond to a generator of order 2, and the number of edges between nodes g and h or a lack thereof should tell me something about. In this work, we will apply our root multiplicity formula to the algebra fa. For a lie algebra l the following are equivalent cf. 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. Affine kacmoody algebras a natural problem is to generalize the theory of.
We present a detailed exposition of the theory of generalized kac moody algebras associated to symmetrizable matrices. Kacmoody lie algebras 9 were introduced in the mid1960s independently by. Jun 29, 2010 w e study representations of the loop kacmoody lie algebra g. Absolutely indecomposable representations and kacmoody lie. It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. Journal of pure and applied algebra elsevier journal of pure and applied algebra 126 1998 233266 generalized kacmoody lie algebras, free lie algebras and the structure of the monster lie algebra elizabeth jurisich department of mathematics, rutgers university new brunswick, nj 08903, usa communicated by c. An exposition of generalized kac moody algebras elizabeth jurisich abstract. Proceedings of the jami inaugural conference, supplement to am. For blocks, note that already in the classical case, nonintegral weights require special. Dimension formula for graded lie algebras 5 multiplicity formula for the roots of level.
David hernandez quantum kacmoody algebras and categori. Generalized kacmoody lie algebras, free lie algebras and the. In this section, we introduce the simplest example of a kacmoody algebra, called af. A systematic study of kacmoody algebras was started independently by v. Local lie algebras, kacmoody algebras, integrable highest weight modules, affine algebras. Integral intertwining operators and complex powers of differential. The lie algebra, being a linear object, is more immediately accessible than the. A finitedimensional lie algebra is semisimple if its solvable radical is 0. Concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations, this is the third revision of an important monograph. Vertex algebras and whittaker modules for affine lie algebras. Before going to my concerned topic kacmoody algebra, we need to get a precise definition of lie algebra, i. Weilbott picture of the cohomology of line bundles over the flag manifold. From the data of the kacmoody lie algebra and a choice.
Let l ct, t1 be the ring of laurent polynomials in t, and set lb l oc 0. Gauge theories of kacmoody extensions w algebras as. 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. 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. The generators of the sun kacmoody extension of the areapreserving di. This lie algebra vdv contains the kacmoody algebra a as a subalgebra but is always far larger than a. 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. Recall that the lie algebra sl 2 is the lie algebra of 2 2 complex matrices with vanishing trace. Kacmoody algebra embedded into g in a canonical manner. 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. I am reading about infinite dimensional lie algebras from the books infinitedimensional lie algebras by victor g. Let hn be the lie algebra which as a vector space has the base ai. I have learned that given a dynkin diagram corresponding to a kac moody algebra, i should be able to use the diagram to read off the generators and relations of the weyl group of that algebra.
This lie algebra vdv contains the kac moody algebra a as a subalgebra but is always far larger than a. These are the notes for a part iii course given in the university of cambridge in autumn 1998. Kacmoody groups as discrete groups bertrand remy cnrs. An approach through invariants and representations, by. X for the flag manifolds x gb g being a simple algebraic. A completion of the quantized enveloping algebra of a kac. This result is a generalization of the borelweil theorem for the flag manifold of a kacmoody algebra, which gives realizations of a finite dimensional irreducible representation of the algebra. Lies seminal idea was to look at the action infinitesimally. Frobenius splitting of thick flag manifolds of kacmoody algebras. If youre reading this essay far in the future because youre interested in the topic i. A kacmoody algebra may be defined by first giving the following.
A detailed treatment of the lie algebra aspect of the theory can be found in v. 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. Kac moody lie algebras as bps lie algebras youtube. Examples of orbit lie algebras arising from 2fold affinization and lorentzian kacmoody algebras are discussed. Each chapter begins with a motivating discussion and ends with a collection of exercises with. Journal of pure and applied algebra elsevier journal of pure and applied algebra 126 1998 233266 generalized kac moody lie algebras, free lie algebras and the structure of the monster lie algebra elizabeth jurisich department of mathematics, rutgers university new brunswick, nj 08903, usa communicated by c. In chapter1, i have given the definitions of lie algebra and some of the terms related to lie algebra, i. Standard references are the book of kac kac, the books of wakimoto wak1,wak2, the survey. The idea is to study the structure of each homogeneous space ln as a module over the affine kacmoody lie algebra lo. Highest weight modules over the lie algebra ga pages 103118.
The simplest example is the one where the local lie group acts on itself by leftor right translations and we get the lie algebra of the lie group. In mathematics, an affine lie algebra is an infinitedimensional lie algebra that is constructed in a canonical fashion out of a finitedimensional simple lie algebra. A conjecture of kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated kacmoody lie algebra. W e study representations of the loop kacmoody lie algebra g. The geometry of flag varieties of a lie algebra g is ubiquitous in representation theory. Moody, and subsequently many results of the theory of finitedimensional semisimple lie algebras have been carried over to kacmoody algebras. They naturally generalise finite dimensional semisimple lie algebras. In the finite case, 9 is nothing but a semisimple y simply connected algebraic group and x is the flag variety 9. The higher kacmoody analogs of the exceptional semisimple lie algebras e7, e7, e8 are. Some subalgebras of indefinite type kacmoody lie algebras. Vertex algebras, kacmoody algebras, and the monster. Part of the progress in mathematics book series pm, volume 204.
Kacmoody lie algebras where simultaneously introduced by kac and moody in the 1960s. The idea is to study the structure of each homogeneous space ln as a module over the affine kac moody lie algebra lo. Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181. These considerations will be recurrent in the following. An exposition of generalized kacmoody algebras elizabeth jurisich abstract. The flag manifold of kacmoody lie algebra, ps 21kb pdf 38kb. I have learned that given a dynkin diagram corresponding to a kacmoody algebra, i should be able to use the diagram to read off the generators and relations of the weyl group of that algebra.
Over c there are several equivalent definitions of semisimple lie algebra. See 3,5 for further background on kacmoody algebras. As a gmodule, v kg can be realized as a generalized verma module. In this paper we prove these conjectures for indivisible.
Oct 16, 2003 a conjecture of kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated kac moody lie algebra. Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems. In the general case we get a lie algebra of vector. Finally, inchapter 3, i explain how the 2kacmoody algebra categori. Kac and lie algebras of finite and affine type by roger carter.
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