Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras
Dmitry Fuchs, Constance Wilmarth
备注：
This is a contribution to the Special Issue on Kac-Moody Algebras and Applications, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of $sl_{2}$ (Theorem 3). The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986), no. 2, 103-113]. In the simpler case of $A_{1}^{1}$ the formula was obtained in [Fuchs D., Funct. Anal. Appl. 23 (1989), no. 2, 154-156].
We consider a general parabolic category $\mathcal{O}_{S}$ over symmetrizable Kac-Moody algebras. We introduce a reduction rule of hom-spaces between parabolic Verma modules over different Kac-Moody algebras which yield some applications on parabolic Verma module homomorphisms.
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We consider epimorphisms from quantum minimal surface algebras onto involutroy subalgebras of split real simply-laced Kac-Moody algebras and provide examples of affine and finite type. We also provide epimorphisms onto such Kac-Moody algebras themselves, where reality of the construction is important. The results extend to the complex situation.
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Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. We introduce principal series representations for these algebras. We study these representations and partially generalize Kato and Matsumoto irreducibility criteria.
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It is shown that any generalized Kac-Moody Lie algebra g that has no mutually orthogonal imaginary simple roots can be written as the vector space direct sum of a Kac-Moody subalgebra and subalgebras isomorphic to free Lie algebras over certain modules for the Kac-Moody subalgebra. Also included is a detailed discussion of Borcherds' construction of the Monster Lie algebra from a vertex algebra and an elementary proof of Borcherds' theorem relating Lie algebras with `an almost positive definite bilinear form' to generalized Kac-Moody algebras. (Preprint version 1996)
The Kac-Moody correction factor, first studied by Macdonald in the affine case, corrects the failure of an identity found by Macdonald in finite-dimensional root systems in 1972. Subsequntly this factor appeared in several formulas in the affine or Kac-Moody analogue of $p$-adic spherical theory for reductive groups. In this article we view the inverse of this correction factor as a function, prove the convergency and holomorphy of this function on a certain domain.
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There exist principal $\mathfrak{sl}_2$ subalgebras for hyperbolic Kac-Moody Lie algebras. In the case of rank 2 symmetric hyperbolic Kac-Moody Lie algebras, certain $\mathfrak{sl}_2$ subalgebras are constructed. These subalgebras are generalizations of principal $\mathfrak{sl}_2$ subalgebras. We show that the rank 2 symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly decomposed under the action of this $\mathfrak{sl}_2$ subalgebras. Furthermore, we classify irreducible components of the decomposition. In particular, we obtain multiplicities of unitary principal series and c…
We introduce a generalization of the classical Hall-Littlewood and Kostka-Foulkes polynomials to all symmetrizable Kac-Moody algebras. We prove that these Kostka-Foulkes polynomials coincide with the natural generalization of Lusztig's $t$-analog of weight multiplicities, thereby extending a theorem of Kato. For $g$ an affine Kac-Moody algebra, we define $t$-analogs of string functions and use Cherednik's constant term identities to derive explicit product expressions for them.
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The restricted category O at the critical level over an affine Kac-Moody algebra is a certain subcategory of the ordinary BGG-category O. We study a deformed version introduced by Arakawa and Fiebig and calculate the center of the deformed restricted category O.
A correction factor naturally arises in the theory of p-adic Kac--Moody groups. In this paper, we expand the correction factor into a sum of irreducible characters of the underlying Kac--Moody algebra. We derive a formula for the coefficients, which lie in the ring of power series with integral coefficients. In the case that the Weyl group is a universal Coxeter group, we show that the coefficients are actually polynomials.
We determine the Lie subalgebra $\mathfrak{g}_{nil}$ of a Borcherds symmetrizable generalized Kac-Moody Lie algebra $\mathfrak{g}$ generated by $ad$-locally nilpotent elements and show that it is `essentially' the same as the Levi subalgebra of $\mathfrak{g}$ with its simple roots precisely the real simple roots of $\mathfrak{g}$.
Generalized root systems (GRS), that were introduced by V. Serganova, are a generalization of finite root systems (RS). We define a generalization of affine root systems (ARS), which we call $\textit{affine generalized root systems}$ (AGRS). The set of real roots of almost every symmetrizable affine indecomposable Kac-Moody superalgebra is an irreducible AGRS. In this paper we classify all AGRSs and show that almost every irreducible AGRS is the set of real roots of a symmetrizable affine indecomposable Kac-Moody superalgebra.
We study Lusztig's t-analog of weight multiplicities associated to level one representations of twisted affine Kac-Moody algebras. An explicit closed form expression is obtained for the corresponding t-string function using constant term identities of Macdonald and Cherednik. The closed form involves the generalized exponents of the graded pieces of the twisted affine algebra, considered as modules for the underlying finite dimensional simple Lie algebra. This extends previous work on level 1 t-string functions for the untwisted simply-laced affine Kac-Moody algebras.
Weyl modules were originally defined for affine Lie algebras by Chari and Pressley in \cite{CP}. In this paper we extend the notion of Weyl modules for a Lie algebra $\mathfrak{g} \otimes A$, where $\mathfrak{g}$ is any Kac-Moody algebra and A is any finitely generated commutative associative algebra with unit over $\mathbb{C}$, and prove a tensor product decomposition theorem generalizing \cite{CP}.
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The exceptional infinite-dimensional linearly compact simple Lie superalgebra ${E}(5,10)$, which Kac believes, is the algebra of symmetries of the ${SU}_{5}$ Grand Unified Model. In this paper, we give a proof of Kac and Rudakov's conjecture about the classification of all the degenerate generalized Verma module over ${E}(5,10)$. Also, we work out all the nontrivial singular vectors degree by degree. It is a potential that the representation theory of ${E}(5,10)$ will shed new light on various features of the the ${SU}_{5}$ Grand unified model.
We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of C-admissible pair as introduced by H. Rubenthaler and J. Nervi for semi-simple and affine Lie algebras. If g is a Kac-Moody Lie algebra (with Dynkin diagram indexed by I) and (I,J) is such a C-admissible pair, we construct a C-admissible subalgebra g^J, which is a Kac-Moody Lie algebra of the same type as g, and whose root system Σgrades finitely the Lie algebra g. For an admissible quotient ρ: I \rightarrow I we build also…
In this paper we have computed all the affine Kac-Moody symmetric spaces which are tame Frechet manifolds starting from the Vogan diagrams related to the affine untwisted Kac-Moody algebras. The detail computation of affine Kac-Moody symmetric spaces associated with A_1^(1) and A_2^(1) are shown algebraically to corroborate our method.
In this paper, we give polyhedral realization of the crystal $B(\infty)$ of $U_q^-(\mathfrak g)$ for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of rank 2, 3 and {\it Monster Lie algebras}.
Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues known as Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side: More precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and desc…
In this paper we prove the Kazhdan-Lusztig type character formula for irreducible highest weight modules with positive rational highest weights over symmetrizable Kac-Moody Lie algebras.
For a finite-dimensional semisimple Lie algebra $\mathfrak{g}$, the Jacobson--Morozov theorem gives a construction of subalgebras $\mathfrak{sl}_2 \subset \mathfrak{g}$ corresponding to nilpotent elements of $\mathfrak{g}$. In this note, we propose an extension of the Jacobson--Morozov theorem to the symmetrizable Kac--Moody setting and give a proof of this generalization in the case of rank two hyperbolic Kac--Moody algebras. We also give a proof for an arbitrary symmetrizable Kac--Moody algebra under some stronger restrictions.
In this paper we give top most and least indices of non vanishing cohomology modules of line bundles in the Kac Moody setting in terms of Weyl group combinatorics. We also prove a surjection Theorem.
Given an affine Kac-Moody algebra, we explain how to construct a coproduct for its associated Yangian. In order to prove that this coproduct is an algebra homomorphism, we obtain, in the first half of this paper, a minimalistic presentation of the Yangian when the Kac-Moody algebra is, more generally, symmetrizable.
By a generalized Tannaka-Krein reconstruction we associate to the admissible representations of the category O of a Kac-Moody algebra, and its category of admissible duals a monoid with a coordinate ring. The Kac-Moody group is the Zariski open dense unit group of this monoid. The restriction of the coordinate ring to the Kac-Moody group is the algebra of strongly regular functions introduced by Kac and Peterson. This monoid has similar structural properties as a reductive algebraic monoid. In particular it is unit regular, its idempotents related to the faces of the Tits cone. It has Bruhat a…
We introduce generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka.
We provide a geometric realization of the crystal $B(\infty)$ for quantum generalized Kac-Moody algebras in terms of the irreducible components of certain Lagrangian subvarieties in the representation spaces of a quiver.
We described in [M1] a monoid acting on the integrable highest weight modules of a symmetrizable Kac-Moody algebra. It has similar structural properties as a reductive algebraic monoid with unit group a Kac-Moody group. Now we find natural extensions of the action of the Kac-Moody group on its building to actions of this monoid. These extensions are partly motivated by representation theory and the combinatorics of the faces of the Tits cone.
We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the Langlands dual affine Kac-Moody group. In the next publication we shall use these results to define and study the notion of Hecke eigenfunction for the group $G_{\aff}$
Using vertex operators, we build representations of the Yangian of a simply laced Kac-Moody algebra and of its double. As a corollary, we prove the PBW property for simply laced affine Yangians.
The aim of this paper is to extend the theory of standard subalgebras of finite dimensional simple Lie algebras to infinite dimensional Lie algebras. We construct and characterize a class of standard subalgebras of affine Kac-Moody algebra.
In 1980 Lusztig proved a stabilisation property of the affine Kazhdan-Lusztig polynomials. In this paper we give a categorical version of such a result using the theory of sheaves on moment graphs. This leads us to associate with any Kac-Moody algebra its stable moment graph.
In this paper, we introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and study their fundamental properties. We then prove the crystal embedding theorem and give a characterization of the crystals $B(\infty)$ and $B(\la)$.
For a dominant integral weight $lambda $, we introduce a family of $U_q ^+ (mathfrak{g})$-submodules $V_w (lambda)$ of the irreducible highest weight $U_q (mathfrak{g})$-module $V(lambda)$ of highest weight $lambda $ for a generalized Kac--Moody algebra $mathfrak{g}$. We prove that the module $V_w (lambda)$ is spanned by its global basis, and then give a character formula for $V_w (lambda)$, which generalizes the Demazure character formula for ordinary Kac--Moody algebras.
In this paper we discuss the isomorphism types of parabolic subgroups in Kac-Moody groups. The results have applications in the study of topology of Kac-Moody groups and their classifying spaces.
