Irreducible Highest Weight Modules Research Articles

Weyl–Kac type character formula for admissible representations of Borcherds–Kac–Moody Lie superalgebras

We prove that Borcherds–Kac–Moody Lie superalgebras yield Weyl–Kac type character formulas for some specific types of admissible representations. Our results generalize the results of Kac and Wakimoto (Proc Natl Acad Sci USA 85:4956–4960, 1988) by extending the Weyl–Kac character formula for integrable irreducible highest weight modules over Kac–Moody algebras to the case of admissible irreducible highest weight modules over Borcherds–Kac–Moody Lie superalgebras.

Representations of [formula omitted]-orbifold of the parafermion vertex operator algebra K(sl2,k)

In this paper, the irreducible modules for the Z2-orbifold vertex operator subalgebra of the parafermion vertex operator algebra associated to the irreducible highest weight modules for the affine Kac-Moody algebra A1(1) of level k are classified and constructed.

Contravariant Form on Tensor Product of Highest Weight Modules

We give a criterion for complete reducibility of tensor product $V\otimes Z$ of two irreducible highest weight modules $V$ and $Z$ over a classical or quantum semi-simple group in terms of a contravariant symmetric bilinear form on $V\otimes Z$. This form is the product of the canonical contravariant forms on $V$ and $Z$. Then $V\otimes Z$ is completely reducible if and only if the form is non-degenerate when restricted to the sum of all highest weight submodules in $V\otimes Z$ or equivalently to the span of singular vectors.

Cayley transform and generalized Whittaker models for irreducible highest weight modules

Cayley transform and generalized Whittaker models for irreducible highest weight modules

Non-weight modules over algebras related to the Virasoro algebra

In this paper, we study some non-weight modules over the loop-Virasoro algebra L and Block type Lie algebras B(q), where q is a nonzero complex number. Those modules when restricted to the Cartan subalgebra (modulo center) are free of rank one. We provide a sufficient and necessary condition for such modules to be simple, and determine their isomorphism classes. Moreover, we obtain the simplicity of modules over loop-Virasoro algebras by taking tensor products of some irreducible modules mentioned above with irreducible highest weight modules or Whittaker modules.

Degenerate flag varieties and Schubert varieties: a characteristic free approach

We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.

Gel’fand–Zetlin basis for a class of representations of the Lie superalgebra

A new, so called odd Gel’fand–Zetlin (GZ) basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra . The related GZ patterns are based upon the decomposition according to a particular chain of subalgebras of . This chain contains only genuine Lie superalgebras of type with k and l nonzero (apart from the final element of the chain which is ). Explicit expressions for a set of generators of the algebra on this GZ basis are determined. The results are extended to an explicit construction of a class of irreducible highest weight modules of the general linear Lie superalgebra .

The Hall module of an exact category with duality

We construct from a finitary exact category with duality A a module over its Hall algebra, called the Hall module, encoding the first order self-dual extension structure of A. We study in detail Hall modules arising from the representation theory of a quiver with involution. In this case we show that the Hall module is naturally a module over the specialized reduced σ-analogue of the quantum Kac–Moody algebra attached to the quiver. For finite type quivers, we explicitly determine the decomposition of the Hall module into irreducible highest weight modules.

Quantum dimensions and fusion rules for parafermion vertex operator algebras

The quantum dimensions and the fusion rules for the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A_1^{(1)} of level k are determined.

Conditioned random walks from Kac-Moody root systems

Random paths are time continuous interpolations of random walks. By using Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra g a random path W. Under suitable hypotheses, we make explicit the probability of the event E: W never exits the Weyl chamber of g. We then give the law of the random walk defined by W conditioned by the event E and proves this law can be recovered by applying to W the generalized Pitmann transform introduced by Biane, Bougerol and O'Connell. This generalizes the main results of [10] and [16] to Kac Moody root systems and arbitrary highest weight modules. Moreover, we use here a completely new approach by exploiting the symmetry of our construction under the action of the Weyl group of g rather than renewal theory and Doob's theorem on Martin kernels.

Irreducible Virasoro modules from tensor products

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega(\lambda,b)$ defined in [LZ], with irreducible highest weight modules $V(\theta,h)$ or with irreducible Virasoro modules Ind$_{\theta}(N)$ defined in [MZ2]. We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $\Omega(\lambda,b)$ with a classical Whittaker module is isomorphic to the module $\mathrm{Ind}_{\theta,\lambda}(\mathbb{C_\mathbf{m}})$ defined in [MW]. As a by-product we obtain the necessary and sufficient conditions for the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to be irreducible. We also generalize the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $ n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are studied and an open problem in [GLZ] is solved. Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.

Affine Lie algebras and conditioned space-time Brownian motions in affine Weyl chambers

We construct a sequence of Markov processes on the set of dominant weights of an affine Lie algebra $\mathfrak{g}$ considering tensor product of irreducible highest weight modules of $\mathfrak{g}$ and specializations of the characters involving the Weyl vector $\rho$. We show that it converges towards a space-time Brownian motion with a drift, conditioned to remain in a Weyl chamber associated to the root system of $\mathfrak{g}$.

Characters of (relatively) integrable modules over affine Lie superalgebras

In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules L over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras \({\mathfrak{g}}\). The problem consists of two parts. First, it is the reduction of the problem to the \({\overline{\mathfrak{g}}}\)-module F(L), where \({\overline{\mathfrak{g}}}\) is the associated to L integral Lie superalgebra and F(L) is an integrable irreducible highest weight \({\overline{\mathfrak{g}}}\)-module. Second, it is the computation of characters of integrable highest weight modules. There is a general conjecture concerning the first part, which we check in many cases. As for the second part, we prove in many cases the KW-character formula, provided that the KW-condition holds, including almost all finite-dimensional \({\mathfrak{g}}\)-modules when \({\mathfrak{g}}\) is basic, and all maximally atypical non-critical integrable \({\mathfrak{g}}\)-modules when \({\mathfrak{g}}\) is affine with non-zero dual Coxeter number.

Automorphism group of parafermion vertex operator algebras

The automorphism group of parafermion vertex operator algebra associated with the irreducible highest weight module for the affine Kac–Moody algebra A1(1) was easily determined since the Virasoro primary vector of weight 3 in this parafermion vertex operator algebra is unique up to a scalar. However, it is highly nontrivial to determine the automorphism group of parafermion vertex operator algebra associated with the irreducible highest weight module for the affine Kac–Moody algebra An(1) with the rank n≥2. As the first step, in this paper, we determine the full automorphism group of parafermion vertex operator algebra associated with the irreducible highest weight module for the affine Kac–Moody algebra A2(1), which shows the idea for a complete determination for the full automorphism group of the parafermion vertex operator algebra associated with the irreducible highest weight module for any affine Kac–Moody algebra.

Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications

We investigate the free field realization of the twisted Heisenberg–Virasoro algebra H at level zero. We completely describe the structure of the associated Fock representations. Using vertex-algebraic methods and screening operators we construct singular vectors in certain Verma modules as Schur polynomials. We completely solve the irreducibility problem for the tensor products of irreducible highest weight modules with intermediate series. We also determine the fusion rules for an interesting subcategory of H-modules. Finally, as an application we present a free-field realization of the W(2,2)-algebra and interpret the W(2,2)-singular vectors as H-singular vectors in Verma modules.

Boson-fermion correspondence of type D-A and multi-local Virasoro representations on the Fock space $\mathit {F^{\otimes \frac{1}{2}}}$F⊗12

We construct the bosonization of the Fock space \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 of a single neutral fermion by using a 2-point local Heisenberg field. We decompose \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 as a direct sum of irreducible highest weight modules for the Heisenberg algebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_{\mathbb {Z}}$\end{document}HZ, and thus we show that under the Heisenberg \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_{\mathbb {Z}}$\end{document}HZ action the Fock space \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 of the single neutral fermion is isomorphic to the Fock space \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes 1}}$\end{document}F⊗1 of a pair of charged free fermions, thereby constructing the boson-fermion correspondence of type D-A. As a corollary we obtain the Jacobi identity equating the graded dimension formulas utilizing both the Heisenberg and the Virasoro gradings on \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12. We construct a family of 2-point-local Virasoro fields with central charge \documentclass[12pt]{minimal}\begin{document}$-2+12\lambda -12\lambda ^2, \ \lambda \in \mathbb {C}$\end{document}−2+12λ−12λ2,λ∈C, on \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12. We construct a W1 + ∞ representation on \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 and show that under this W1 + ∞ action \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 is again isomorphic to \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes 1}}$\end{document}F⊗1.

Irreducible characters of Kac-Moody Lie superalgebras

Generalizing the super duality formalism for finite-dimensional Lie superalgebras of type $ABCD$, we establish an equivalence between parabolic BGG categories of a Kac-Moody Lie superalgebra and a Kac-Moody Lie algebra. The characters for a large family of irreducible highest weight modules over a symmetrizable Kac-Moody Lie superalgebra are then given in terms of Kazhdan-Lusztig polynomials for the first time. We formulate a notion of integrable modules over a symmetrizable Kac-Moody Lie superalgebra via super duality, and show that these integrable modules form a semisimple tensor subcategory, whose Littlewood-Richardson tensor product multiplicities coincide with those in the Kac-Moody algebra setting.

Correction to “Signatures of invariant Hermitian forms on irreducible highest weight modules,” Duke Math. J. 142 (2008), 165–196

Correction to “Signatures of invariant Hermitian forms on irreducible highest weight modules,” Duke Math. J. 142 (2008), 165–196

Relating signed and classical Kazhdan–Lusztig polynomials

Motivated by studying the unitary dual problem, a variation of Kazhdan–Lusztig polynomials was defined in a previous publication by the author which encodes signature information at each level of the Jantzen filtration. These so-called signed Kazhdan–Lusztig polynomials may be used to compute the signatures of invariant Hermitian forms on irreducible highest weight modules. The key result of this paper is a simple relationship between signed Kazhdan–Lusztig polynomials and classical Kazhdan–Lusztig polynomials: signed Kahzdan–Lusztig polynomials are shown to equal classical Kazhdan–Lusztig polynomials evaluated at −q rather than q and multiplied by a sign. A simple signature character inversion formula follows from this relationship. These results have applications to finding the unitary dual for real reductive Lie groups since Harish–Chandra modules may be constructed by applying Zuckerman functors to the highest weight modules.

An algebraic geometric model of an action of the face monoid associated to a Kac–Moody group on its building

The face monoid Gˆ described in [11] acts on the integrable irreducible highest weight modules of a symmetrizable Kac–Moody algebra. It has similar structural properties as a reductive algebraic monoid whose unit group is a symmetrizable Kac–Moody group G. We found in [15] two natural extensions of the action of the Kac–Moody group G on its building Ω to actions of the face monoid Gˆ on the building Ω. Now we give an algebraic geometric model of one of these actions of the face monoid Gˆ on Ω, where the building Ω is obtained as a part of the F-valued points of the spectrum of all homogeneous prime ideals of the Cartan algebra CA of the Kac–Moody group G. We describe the spectrum of all homogeneous prime ideals of the Cartan algebra CA and determine its F-valued points.