Highest Weight Representations Research Articles

String Hypothesis for Spin Chains: A Particle/Hole Democracy

This paper is devoted to integrable gl(n|m) spin chains which allow for formulation of the string hypothesis. Considering the thermodynamic limit of such spin chains, we derive linear functional equations that symmetricaly treat holes and particles. The functional equations naturally organize different types of excitations into a pattern equivalent to the one of Y-system, and, not surprisingly, the Y-system can be easily derived from the functional equations. The Y-system is known to contain most of the information about the symmetry of the model, therefore we map the symmetry knowledge directly to the description of string excitations. Our analysis is applicable for highest weight representations which for some choice of the Kac-Dynkin diagram have only one nonzero Dynkin label. This generalizes known results for the AdS/CFT spectral problem and for the Hubbard model.

一类与Virasoro代数密切相关的W-型李代数的最高权表示

W-代数是用来描述理论物理中的共形场论的一类代数，它的结构理论和表示理论我们都知之甚少。本文研究一类与Virasoro代数密切相关的W型-李代数的最高权表示理论。 W-algebra is a family algebras which are used to describe the conformal field theory. Up to now, the structure theory and representation theory we have obtained are very little. In this paper, we investigate the highest weight representations over a W-type Lie algebra related to Virasoro algebra.

Weyl group multiple Dirichlet series of typeC

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For a root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. They conjecturally arise as Whittaker coefficients of Eisenstein series on a metaplectic group with cover degree n. For type C and n odd, we construct an infinite family of Dirichlet series and prove they satisfy the above analytic properties in many cases. The coefficients are exponential sums built from Gelfand‐Tsetlin bases of certain highest weight representations. Previous attempts to define such series by Brubaker, Bump, and Friedberg required n sufficiently large, so that coefficients were described by Weyl group orbits. We demonstrate that these two radically different descriptions match when both are defined. Moreover, for n D 1, we prove our series are Whittaker coefficients of Eisenstein series on SO.2r C 1/.

Krichever-Novikov algebras, their representations and applications in geometry and mathematical physics

1. Krichever–Novikov algebras and their place in the theory of Lie algebras, geometry and topology of moduli spaces, the theory of integrable systems, and conformal quantum field theory. In the 1960s–1980s, the theory of infinite-dimensional Lie algebras experienced a period of fierce growth related to the appearance of affine Kac–Moody algebras and Virasoro algebras, which combine direct inheritance from the classical theory of semisimple Lie algebras with successful applications. In this period, two important disciplines of mathematical physics were formed, the theory of integrable systems and conformal field theory, characterized by the deep penetration of methods of the theory of Riemann surfaces into theoretical physics. The discovery of the role of Kac–Moody and Virasoro algebras as the algebras of formally gauge and conformal infinitesimal symmetries of these theories and an important tool for integrating nonlinear equations has united both objects. The passage from formal types of symmetries to more geometric ones required considering current algebras and vector fields on Riemann surfaces. In the late 1980s, several works on this topic appeared, which were mainly related to elliptic curves, but not only to them. In 1987, in relation to the study of solitons and conformal field theory, I. M. Krichever and S. P. Novikov introduced the central extensions of current algebras and algebras of vector fields on Riemann surfaces (with complex structure and marked points), which later acquired the name Krichever–Novikov algebras [1–3]. These algebras naturally generalize affine (untwisted) Kac– Moody algebras and Virasoro algebras, respectively. Among other algebras of currents and vector fields on Riemann surfaces the Krichever–Novikov algebras are distinguished by the important property of carrying an almost graded structure, which is weaker than grading but stronger than filtration and leads to many important results; in particular, this makes it possible to consider analogues of highest weight representations. As applied to central extensions, being almost graded is equivalent to the locality property of the corresponding cocycles; moreover, the cocycles are uniquely determined by this property in themost important cases. Affine (untwisted) Kac–Moody algebras andVirasoro algebras are special cases of Krichever–Novikov algebras; from which point of view, they correspond to the Riemann sphere with two marked points. Krichever–Novikov algebras are related in many ways to fundamental problems of geometry, analysis, and mathematical physics. Being introduced as algebras of Fourier–Laurent series on a Riemann surface, they are a part of harmonic analysis. The representation theory of Krichever–Novikov algebras is closely related to the theory of holomorphic bundles on Riemann surfaces. In particular, holomorphic bundles play the key role in parameterizing fermion representations. The this class of representations is most important for the theory among those known at present; it was introduced by the author in [4]. The construction of fermion representations also demonstrates another important relationship, namely, to integrable systems: this construction uses essentially objects of the theory of commutative rings of difference operators constructed in [5–7].

Invariant theory and the $\mathcal{W}_{1+\infty}$ algebra with negative integral central charge

The vertex algebra \mathcal{W}_{1+\infty,c} with central charge c may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n\geq 1 , it was conjectured in the physics literature that \mathcal{W}_{1+\infty,-n} should have a minimal strong generating set consisting of n^2+2n elements. Using a free field realization of \mathcal{W}_{1+\infty,-n} due to Kac-Radul, together with a deformed version of Weyl's first and second fundamental theorems of invariant theory for the standard representation of GL_n , we prove this conjecture. A consequence is that the irreducible, highest-weight representations of \mathcal{W}_{1+\infty,-n} are parametrized by a closed subvariety of \mathbb{C}^{n^2+2n} .

Generalized Castelnuovo–Mumford regularity for affine Kac–Moody algebras

The graded modules over noncommutative algebras often have minimal free resolutions of infinite length, resulting in infinite Castelnuovo–Mumford regularity. In Kang et al. (2010) [6], we introduced a generalized notion of Castelnuovo–Mumford regularity to overcome this difficulty. In this paper, we compute the generalized Castelnuovo–Mumford regularity for integrable highest weight representations of all affine Kac–Moody algebras. It is shown that the generalized regularity depends only on the type and rank of algebras and the level of representations.

On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture

In their recent paper \cite{Alday:2009aq} Alday, Gaiotto and Tachikawa proposed a relation between $\mathcal{N}=2$ four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired from the exact form of instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra $\mathcal{A}=\text{\sf Vir}\otimes\mathcal{H}$ which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of $\mathcal{A}$. The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called $Z_{\text{\sf{bif}}}$ appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in \cite{Alday:2009aq}. We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin-Ono integrable hierarchy.

Cuspidal [formula omitted]-modules and deformations of certain Brauer tree algebras

We show that the algebras describing blocks of the category of cuspidal weight (resp. generalized weight) sl n -modules are one-parameter (resp. multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley–Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and Schur algebras and in the study of the Milnor fiber of Kleinian singularities.

Highest weight representation for Sklyanin algebra sl(3)(u) with application to the Gaudin model

We study the infinite-dimensional Sklyanin algebra sl(3)(u). Specifically we construct the highest weight representation for this algebra in an explicit form. Its application to the Gaudin model is mentioned.

PBW Filtration and Bases for Symplectic Lie Algebras

We study the PBW filtration on the highest weight representations $V(\la)$ of $\msp_{2n}$. This filtration is induced by the standard degree filtration on $U(\n^-)$. We give a description of the associated graded $S(\n^-)$-module $gr V(\la)$ in terms of generators and relations. We also construct a basis of $gr V(\la)$. As an application we derive a graded combinatorial formula for the character of $V(\la)$ and obtain a new class of bases of the modules $V(\la)$.

Highest weight modules and polarized embeddings of shadow spaces

The present paper was inspired by the work on polarized embeddings by Cardinali et al. (J. Algebr. Comb. 25(1):7---23, 2007) although some of our results in it date back to 1999. They study polarized embeddings of certain dual polar spaces, and identify the minimal polarized embeddings for several such geometries. We extend some of their results to arbitrary shadow spaces of spherical buildings, and make a connection to work of Burgoyne, Wong, Verma, and Humphreys on highest weight representations for Chevalley groups. Let Δ be a spherical Moufang building with diagram M over some index set I, whose strongly transitive automorphism group is a Chevalley group $G(\mathbb{F})$ over the field $\mathbb{F}$ . For any non-empty set K?I let Γ be the K-shadow space of Δ. Extending the notion in to this situation, we say that an embedding of Γ is polarized if it induces all singular hyperplanes. Here a singular hyperplane is the collection of points of Γ not opposite to a point of the dual geometry Γ ?, which is the shadow geometry of type opp I (K) opposite to K. We prove a number of results on polarized embeddings, among others the existence of (relatively) minimal polarized embeddings. We assume that $G(\mathbb{F})$ is untwisted. In that case, the point-line geometry Γ has an embedding e K into the Weyl module $V(\lambda_{K})_{\mathbb{F}}^{0}$ of highest weight ? K =? k?K ? k . We show that this embedding is polarized in the sense described above. We then prove that the minimal polarized embedding relative to e K exists and equals the unique irreducible $G(\mathbb{F})$ -module L(? K ) of highest weight ? K . More precisely we show that the polar radical of e K (the intersection of all singular hyperplanes) coincides with the radical of the contravariant bilinear form considered by Wong to obtain the irreducible (restricted) representations of $G(\mathbb{F})$ in positive characteristic. This viewpoint allows us to recognize the irreducible $G(\mathbb{F})$ -modules of highest weight ? K geometrically as minimal polarized embeddings of the appropriate shadow space.

PBW filtration and bases for irreducible modules in type A n

We study the PBW filtration on the highest weight representations V(λ) of \( \mathfrak{s}{\mathfrak{l}_{n + 1}} \). This filtration is induced by the standard degree filtration on \( {\text{U}}\left( {{\mathfrak{n}^{-} }} \right) \). We give a description of the associated graded \( S\left( {{\mathfrak{n}^{-} }} \right) \)-module gr V(λ) in terms of generators and relations. We also construct a basis of gr V(λ). As an application we derive a graded combinatorial character formula for V(λ), and we obtain a new class of bases of the modules V(λ) conjectured by Vinberg in 2005.

High spin limits and non-abelian T-duality

The action of the non-abelian T-dual of the WZW model is related to an appropriate gauged WZW action via a limiting procedure. We extend this type of equivalence to other σ-models with non-abelian isometries and their non-abelian T-duals, focusing on Principal Chiral models. We reinforce and refine this equivalence by arguing that the non-abelian T-duals are the effective backgrounds describing states of an appropriate parent theory corresponding to divergently large highest weight representations. The proof involves carrying out a subtle limiting procedure in the group representations and relating them to appropriate limits in the corresponding backgrounds. We illustrate the general method by providing several non-trivial examples.

Generalized Kac–Moody Lie Algebras and Product Quivers

We construct the entire generalized Kac–Moody Lie algebra as a quotient of the positive part of another generalized Kac–Moody Lie algebra. The positive part of a generalized Kac–Moody Lie algebra can be constructed from representations of quivers using Ringel's Hall algebra construction. Thus we give a direct realization of the entire generalized Kac–Moody Lie algebra. This idea arises from the affine Lie algebra construction and evaluation maps. In [16], we give a quantum version of this construction after analyzing Nakajima's quiver variety construction of integral highest weight representations of the quantized enveloping algebras in terms of the irreducible components of quiver varieties.

Vector coherent state realization of representations of the affine superalgebra A(1)(,1)

The method of vector coherent states is generalized to study the representations of the affine superalgebra A(1)(0,1). A new realization of all the irreducible highest weight representations of this affine superalgebra is obtained in terms of holomorphic superdifferential operators in infinitely many Grassmannian and bosonic variables.

CORWIN–GREENLEAF MULTIPLICITY FUNCTIONS FOR HERMITIAN SYMMETRIC SPACES AND MULTIPLICITY-ONE THEOREM IN THE ORBIT METHOD

Kobayashi's multiplicity-free theorem asserts that irreducible unitary highest-weight representations π are multiplicity-free when restricted to every symmetric pair if π is of scalar type. The aim of this paper is to find the "classical limit" of this multiplicity-free theorem in terms of the geometry of two coadjoint orbits, for which the correspondence is predicted by the Kirillov–Kostant–Duflo orbit method.For this, we study the Corwin–Greenleaf multiplicity function [Formula: see text] for Hermitian symmetric spaces G/K. First, we prove that [Formula: see text] for any G-coadjoint orbit [Formula: see text] and any K-coadjoint orbit [Formula: see text] if [Formula: see text]. Here, 𝔤 = 𝔨 + 𝔭 is the Cartan decomposition of the Lie algebra 𝔤 of G.Second, we find a necessary and sufficient condition for [Formula: see text] by means of strongly orthogonal roots. This criterion may be regarded as the "classical limit" of a special case of the Hua–Kostant–Schmid–Kobayashi branching laws of holomorphic discrete series representations with respect to symmetric pairs.

Kostant homology formulas for oscillator modules of Lie superalgebras

We provide a systematic approach to obtain formulas for characters and Kostant u -homology groups of the oscillator modules of the finite-dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite-dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned u -homology computation are worked out.

AFFINE HALL-LITTLEWOOD FUNCTIONS FOR AFormula AND SOME CONSTANT TERM IDENTITIES OF CHEREDNIK-MACDONALD-MEHTA TYPE

We study t-analogs of string functions for integrable highest weight representations of the affine Kac-Moody algebra A(1)((1)). We obtain closed form formulas for certain t-string functions of levels 2 and 4. As corollaries, we obtain explicit identities for the corresponding affine Hall-Littlewood functions, as well as higher level generalizations of Cherednik's Macdonald and Macdonald-Mehta constant term identities.

Double affine Lie algebras and finite groups

We introduce and begin to study Lie theoretical analogs of symplectic reflection algebras for a finite group, which we call cyclic double affine Lie We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new involves $\mathbb{C}[u,v]$. We describe its universal central extensions and start the study of its representation theory, in particular of its highest weight integrable modules and Weyl modules. We also consider the first Weyl algebra $A_1$ instead of the polynomial ring $\mathbb{C}[u,v]$, and, more generally, a rank one rational Cherednik algebra. We study quasi-finite highest weight representations of these Lie algebras.

Imaginary Highest-Weight Representation Theory and Symmetric Functions

Affine Lie algebras admit nonclassical highest-weight theories through alternative partitions of the root system. Although significant inroads have been made, much of the classical machinery is inapplicable in this broader context, and some fundamental questions remain unanswered. In particular, the structure of the reducible objects in nonclassical theories has not yet been fully understood. This question is addressed here for affine sl(2), which has a unique nonclassical highest-weight theory, termed “imaginary.” The reducible Verma modules in the imaginary theory possess an infinite descending series, with all factors isomorphic to a certain canonically associated module, the structure of which depends upon the highest weight. If the highest weight is nonzero, then this factor module is irreducible, and conversely. This article examines the degeneracy of the factor module of highest-weight zero. The intricate structure of this module is understood via a realization in terms of the symmetric functions. The realization permits the description of a family of singular (critical) vectors and the classification of the irreducible subquotients. The irreducible subquotients are characterized as those modules with an action given in terms of exponential functions, in the sense of Billig and Zhao.