Irreducible Highest Weight Representations Research Articles

Free-field representation of the quantum affine algebra [formula omitted] and form factors in the higher-spin XXZ model

We consider the spin- 1 2 k XXZ model in the antiferromagnetic regime using the free-field realization of the quantum affine algebra U q( sl 2) of level k. We give a free-field realization of the type-II q-vertex operator, which describes creation and annihilation of physical particles in the model. By taking a trace of the type-I and type-II q-vertex operators over the irreducible highest-weight representation of U q( sl 2) , we also derive an integral formula for form factors in this model. Investigating the structure of poles, we obtain a residue formula for form factors, which is a lattice analog of the higher-spin extension of Smirnov's formula in the massive integrable quantum field theory. This result as well as the quantum deformation of the Knizhnik-Zamolodchikov equation for form factors shows a deep connection in the mathematical structure of the integrable lattice models and the massive integrable quantum field theory.

BRST COHOMOLOGY IN QUANTUM AFFINE ALGEBRA $U_q \left( {\widehat{sl_2 }} \right)$

Using free field representation of quantum affine algebra [Formula: see text], we investigate the structure of the Fock modules over [Formula: see text]. The analysis is based on a q-analog of the BRST formalism given by Bernard and Felder in the affine Kac-Moody algebra [Formula: see text]. We give an explicit construction of the singular vectors using the BRST charge. By the same cohomology analysis as the classical case (q=1), we obtain the irreducible highest weight representation space as a non-trivial cohomology group. This enables us to calculate a trace of the q-vertex operators over this space.

Casimir invariants for quantized affine Lie algebras

Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest-weight representation.

Quantum affine Lie algebras, Casimir invariants, and diagonalization of the braid generator

Let Uq(𝒢̂) be an infinite-dimensional quantum affine Lie algebra. A family of central elements or Casimir invariants are constructed and their eigenvalues computed in any integrable irreducible highest weight representation. These eigenvalue formulas are shown to be absolutely convergent when the deformation parameter q is such that ‖q‖≳1. It is proven that the universal R-matrix R of Uq(𝒢̂) satisfies the celebrated conjugation relation R°=TR with T the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight Uq(𝒢̂)-module and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin’s and Gould’s forms to the present affine case. Casimir invariants acting on a specified module are also constructed and their eigenvalues, again absolutely convergent for ‖q‖≳1, computed by means of the spectral decomposition formula.

Eigenvalues of Casimir invariants of Uq (gl(m/n))

A method is investigated for determining the eigenvalues of the Casimir invariants of the quantum supergroup Uq(gl(m‖n)) when acting on irreducible highest weight representations. This leads to a simple eigenvalue formula expressible in terms of the highest weight labels.

Affine Toda solitons and vertex operators

Affine Toda theories with imaginary couplings associate with any simple Lie Algebra g generalisations of sine-Gordon theory which are likewise integrable and posses soliton solutions. The solitons are “created” by exponentials of quantities F ̂ i(z) which lie in the untwisted affine Kac-Moody algebra ĝ and ad-diagonalise the principal Heisenberg subalgebra. When g is simply laced and highest-weight irreducible representations at level one are considered, F ̂ i(z) can be expressed as a vertex operator whose square vanishes. This nilpotency property is extended to all highest-weight representations of all affine untwisted Kac-Moody algebras in the sense that the highest non-vanishing power becomes proportional to the level. As a consequence, the exponential series mentioned terminates and the soliton solutions have a relatively simple algebraic expression whose properties can be studied in a general way. This means that various physical properties of the soliton solutions can be directly related to the algebraic structure. For example, a classical version of Dorey's fusing rule follows from the operator product expansion of two F ̂ ' s , at least when g is simply laced. This adds to the list of resemblances of the solitons with respect to the particles which are the quantum excitations of the fields.

Szegő maps and highest weight representations

Let G be a connected noncompact simple Lie group with finite center and let A be a maximal compact subgroup of G. Assume the space G/K is Hermitian symmetric. We associate to each irreducible representation τ of K a principal series representation W(τ) and a (7-equivariant Szegό-type integral operator Sτ such that Sτ maps the K-finite vectors in W(τ) onto an irreducible highest weight $module L(τ). Of primary concern here are those representations τ which are reduction points. For such τ, we construct certain systems 3lτ of ΰ-equivariant differential operators and then utilize 3fτ to establish the infinitesimal irreducibility of the image of Sτ. 1. Introduction. Let G be a connected noncompact simple Lie group with finite center and let K be a maximal compact subgroup of G. Assume the space G/K is Hermitian symmetric. The main purpose of this article is to realize each irreducible highest weight representation of G as the image of a G-equivariant quotient map defined on principal series representations. To make this more precise, recall that each irreducible highest weight representation πτ of G is parametrized by an irreducible unitary representation τ of K. Let C°°(G, τ) denote the space of τ-covariant C°°-functions on G. We associate to τ a principal series representation W(τ) and a Szego map Sτ: W(τ) -> C°°(G, τ) having the property that the ^-finite vectors in W(τ) are mapped onto an irreducible g-module equivalent to the derived action of πτ. In the case of discrete series and limits thereof, this type of result was proved by Knapp and Wallach [16] in the general setting where G is a semisimple equirank Lie group with finite center. The main result here is that the irreducibility of the image of *S τ persists for all highest weight representations. Moreover, for certain τ called reduction points, the irreducibility of Image (Sτ) is proved by showing this space is annihilated by a system Qίτ of Gequivariant differential operators. The system 3τ somewhat parallels the role of the Schmid operator in the Knapp and Wallach result. The realization of distinguished representations as irreducible images of quotient maps is a recurring theme in the literature which

Finite dimensional irreducible representations of the quantum supergroup Uq (gl(m/n))

Finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n)) at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction. The representation theory is rather similar to that of gl(m/n) at generic q, but drastically different when q is a root of unity. In the latter case, atypicality conditions of highest weight irreducible representations (irreps) are substantially altered, and such finite-dimensional irreps arise that do not have highest weight and/or lowest weight vectors. As concrete examples, the irreps of Uq(gl(2/1)) are classified.

Oscillator realization of the strange superalgebras P(n) and Q(n)

The strange superalgebra P(n) is realized in terms of bilinear products of oscillators of bosonic and fermionic type. The even part is given by sums of bilinears in bosons and fermions, and the odd part by bilinears in one boson and one fermion. These oscillators can be used to build explicit realizations of some highest weight irreducible representations of P(n). This procedure is illustrated for P(3), where, in particular, the case of the adjoint representation of P(3) is worked out in detail. Finally, an oscillator realization of the strange superalgebra Q(n) is also given.

Finite W-algebras

Finite versions of W-algebras are introduced by considering (symplectic) reductions of finite dimensional simple Lie algebras. It is shown that a nonlinear su(2) algebra previously introduced by Roček [Phys. Lett. B 255 (1991) 554] is nothing but a finite analogue of W 3 (2) and therefore a reduction of sl 3. Its unitary and non-unitary, reducible and irreducible highest weight representations are constructed.

Irreducible highest weight representations of quantum groupsU q (gl(n, C))

Explicit recurrence formulas of canonical realization (boson representation) for quantum enveloping algebrasUq(gl(n, C)) are given. Using them, irreducible highest weight representations ofUq(gl(n, C)) are obtained as restriction of representation of Fock space to invariant subspace generated by vacuum as a cyclic vector.

AN EXPLICIT FORMULA FOR SOME SINGULAR VECTORS OF THE NEVEU–SCHWARZ ALGEBRA

Similarly to the Virasoro algebra, the Neveu–Schwarz algebra has a discrete series of unitary irreducible highest weight representations. These are labeled by the values of [Formula: see text] (the central charge) and of the highest weight hpq = [(p (m + 2) − qm)2 − 4]/(8m (m + 2)) where m, p, q are some integers. The Verma modules constructed with these values (c, h) are not irreducible, however, as they contain two Verma submodules, each generated by a singular vector ψp,q (of weight hpq + pq/2) and ψm−p, m+2−q (of weight hpq + (m−p)(m+2−q)/2), respectively. We give an explicit expression for these singular vectors whenever one of its indices is 1.

BERNSTEIN-GEL’FAND-GEL’FAND RESOLUTION OF W3 ALGEBRA IN THE LATTICE APPROACH

We describe the embedding structure of the Verma modules of the W3 algebra using the free field representation (FFR) in the lattice approach. This structure is expressed by a set of intertwining diagrams. In particular, we show how these diagrams can be used to achieve the Bernstein-Gel’fand-Gel’fand (BGG) resolution of the irreducible highest weight representation (IHWR) in terms of the Verma modules. As an application of prime importance, we show how the character of the W3 IHWR can be readily derived using the Rocha-Caridi procedure.

Orbits and characters associated to highest weight representations

We relate two different orbit decompositions of the flag variety. This allows us to pass from the closed formulas of Boe, Enright, and Shelton for the formal character of an irreducible highest weight representation to closed formulas for the distributional character written as a sum of characters of generalized principal series representations. Otherwise put, we give a dictionary between certain Lusztig-Vogan polynomials arising in Harish-Chandra module theory and the Kazhdan-Lusztig polynomials associated to a relative category O \mathcal {O} of Hermitian symmetric type.

Universal R matrices and invariants of quantum supergroups

A general method is developed for constructing invariants of quantum supergroups using universal R matrices. Applied to Uq(gl(2/1)), this method yields the invariants of this quantum supergroup in explicit form. The eigenvalues of these invariants in arbitrary irreducible highest weight representations are computed.

Dynkin-like diagrams and representations of the strange superalgebra P(n)

Using an extension P̃(n) of the noncontragredient superalgebra P(n), on which a nondegenerate Killing form can be constructed, simple root systems (SRSs) are introduced. To each SRS is associated a Dynkin-like diagram. Then the maximal regular sub-(super)algebras are obtained. Moreover, a procedure to construct highest weight irreducible representations of P(n) is presented.

Twisted vertex representations of quantum affine algebras

Recent interests in quantum groups are stimulated by their marvelous relations with quantum Yang-Baxter equations, conformal field theory, invariants of links and knots, and q-hypergeometric series. Besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q-deformations or quantum analogues of known structures. Quantum groups were first defined by Drinfeld [2] and Jimbo [9] (also see [4]) as a q-deformation of the universal enveloping algebras of the KacMoody algebras in the work of trigonometric solutions of Yang-Baxter equations. In the same spirit it was shown in [13], [14], that there exists a 1 1 correspondence between the integrable highest weight representations of symmetrizable Kac-Moody algebras and those of the corresponding quantum groups, where both spaces have the same dimension in the case of generic q (i.e. q is not a root of unity). Moreover, one can be very explicit in the case of quantum gl(n) to write down the irreducible highest weight representations. Quantum affine algebras are the quantum groups associated to affine Lie algebras. Following Drinfeld's realization [-3] of q-analog of loop algebras, the vertex representation of untwisted simply laced quantum affine algebras was constructed in Frenkel-Jing [6], which is a q-deformation of Frenkel-Kac [7] and Segal [15] construction in the theory of affine Lie algebras. Subsequently, the same was done for the quantum affine algebra of type B in [1]. In the present work we construct vertex representations of quantum affine algebras twisted by an automorphism of the Dynkin diagram, which generalizes certain important cases in the ordinary twisted vertex operator calculus [-5,

The partition function of the 2D Ising model with magnetic fields on the boundaries and [formula omitted] Virasoro characters

The partition function of a planar Ising model on a finite lattice with magnetic fields on the boundaries is represented through an anticommuting functional integral with Gaussian distribution. In particular, the previously unknown solution for the case of fields of opposite direction is obtained. It is shown also that the partition of the model at the critical point in the continuum limit can be expressed through certain characters of the highest weight irreducible representations of the Virasoro algebra with c= 1 2 .

A coset construction for the super W 3 algebra

An explicit realization of the super W 3 (SW 3) algebra is performed using the coset construction (SU(3) 3 ⊗ SU(3) 1)/SU(3) 4. The corresponding value of c=10/7 is the only one for which unitary representations of the SW 3 exist. We then determine the unitary highest-weight irreducible representation series of this algebra.

Highest weight irreducible unitarizable representations of Lie algebras of infinite matrices. The algebra A∞

In this paper all unitarizable irreducible highest weight representations of the infinite-dimensional Lie algebra A∞, which is a completion and central extension of the general linear Lie algebra gl∞, are considered. Within each representation space a basis is introduced and expressions for the transformations of the basis under the action of the Chevalley generators are written.