Highest Weight Modules Research Articles

Biadjointness in Cyclotomic Khovanov–Lauda–Rouquier Algebras

In this paper, we prove that a pair of functors \mathsf E_{1}^{\Lambda} and \mathsf F_{1}^{\Lambda} appearing in the categorification of irreducible highest weight modules of quantum groups via cyclotomic Khovanov-Lauda-Rouquier algebras is a biadjoint pair.

Certain categories of modules for twisted affine Lie algebras

In this paper, using generating functions, we study two categories Open image in new window and Open image in new window of modules for twisted affine Lie algebras Open image in new window , which were firstly introduced and studied for untwisted affine Lie algebras by H. -S. Li [Math Z, 2004, 248: 635–664]. We classify integrable irreducible Open image in new window -modules in categories Open image in new window and Open image in new window , where Open image in new window is proved to contain the well-known evaluation modules and Open image in new window to unify highest weight modules, evaluation modules and their tensor product modules. We determine also the isomorphism classes of those irreducible modules.

Irreducible highest weight representations of the simple n-Lie algebra

A. Dzhumadil'daev classified all irreducible finite dimensional representations of the simple n-Lie algebra. Using a slightly different approach, we obtain in this paper a complete classification of all irreducible, highest weight modules, including the infinite-dimensional ones. As a corollary we find all primitive ideals of the universal enveloping algebra of this simple n-Lie algebra.

On Projective Modules in Category $\mathcal{O}_{int}$ of Quantum $\mathfrak{sl}_{2}$

The restriction of a Verma module of \({\bf U}(\mathfrak{sl}_3)\) to \({\bf U}(\mathfrak{sl}_2)\) is isomorphic to a Verma module tensoring with all the finite dimensional simple modules of \({\bf U}(\mathfrak{sl}_2)\). The canonical basis of the Verma module is compatible with such a decomposition. An explicit decomposition of the tensor product of the Verma module of highest weight 0 with a finite dimensional simple module into indecomposable projective modules in the category \(\mathcal O_{\rm{int}}\) of quantum \(\mathfrak{sl}_2\) is given.

Classification of Simple Weight Modules for the Neveu–Schwarz Algebra with a Finite-Dimensional Weight Space

We show that the support of a simple weight module over the Neveu–Schwarz algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such module are infinite-dimensional. As a corollary we obtain that every simple weight module over the Neveu–Schwarz algebra, having a nontrivial finite-dimensional weight space, is a Harish–Chandra module (and hence is either a highest or lowest weight module, or else a module of the intermediate series). This result generalizes a theorem which was originally given on the Virasoro algebra.

Quantum Sphere $${\mathbb{S}^4}$$ as a Non-Levi Conjugacy Class

We construct a U_h(sp(4))-equivariant quantization of the four-dimensional complex sphere S^4 regarded as a conjugacy class, Sp(4)/Sp(2)x Sp(2), of a simple complex group with non-Levi isotropy subgroup, through an operator realization of the quantum polynomial algebra C_h[S^4] on a highest weight module of U_h(sp(4)).

θ-Unitary representations for the W-algebra W(2,2)

In this article, conjugate-linear anti-involutions and θ-unitary Harish-Chandra modules over the W-algebra W(2,2) are studied. It is proved that there are only two classes that conjugate-linear anti-involutions over W(2,2). The main result of this article is that a θ-unitary Harish-Chandra module over the W-algebra W(2,2) is simply a -unitary Harish-Chandra module over the Virasoro algebra and hence is either a simple highest or lowest weight module or a simple module from the intermediate series of the form A a,b for some

Littlewood identity and crystal bases

We give a new combinatorial model for the crystals of integrable highest weight modules over the classical Lie algebras of type B and C in terms of classical Young tableaux. We then obtain a new description of its Littlewood–Richardson rule and a maximal Levi branching rule in terms of classical Littlewood–Richardson tableaux, which extends in a bijective way the well-known stable formulas at large ranks. We also show that this tableau model admits a natural superization and it produces the characters of irreducible highest weight modules over orthosymplectic Lie superalgebras, which correspond to the integrable highest weight modules over the classical Lie algebras of type B and C under the Cheng–Lam–Wangʼs super duality.

Dirac cohomology of highest weight modules

We show that the Dirac cohomology coincides with nilpotent Lie algebra (co)homology up to a twist of a one-dimensional character for simple highest weight modules. As a consequence, we determine the Dirac cohomology of simple highest weight modules explicitly.

A Bott–Borel–Weil Theorem for Diagonal Ind-groups

AbstractA diagonal ind-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusionas a typical special case. If G is a diagonal ind-group and B ⊂ G is a Borel ind-subgroup, we consider the ind-variety G/B and compute the cohomology H𝓁(G/B,𝒪−λ) of any G-equivariant line bundle 𝒪−λ on G/B. It has been known that, for a generic λ, all cohomology groups of 𝒪−λ vanish, and that a non-generic equivariant line bundle 𝒪−λ has at most one nonzero cohomology group. The new result of this paper is a precise description of when Hj (G/B,𝒪−λ) is nonzero and the proof of the fact that, whenever nonzero, Hj (G/B,𝒪−λ) is a G-module dual to a highest weight module. The main difficulty is in defining an appropriate analog WB of the Weyl group, so that the action of WB on weights of G is compatible with the analog of the Demazure “action” of the Weyl group on the cohomology of line bundles. The highest weight corresponding to Hj (G/B,𝒪−λ) is then computed by a procedure similar to that in the classical Bott–Borel–Weil theorem.

A chiral Borel–Weil–Bott theorem

We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. This is applied to (1) finding the character of G-integrable irreducible highest weight modules over the affine Lie algebra at the critical level, and (2) computing a certain elliptic genus of the flag manifold. The main tool is a result that interprets the Drinfeld–Sokolov reduction as a derived functor.

Quasifinite representations of a class of Block type Lie algebras [formula omitted

Intrigued by a well-known theorem of Mathieu’s on Harish-Chandra modules over the Virasoro algebra, we show that any quasifinite irreducible module over a class of Block type Lie algebras B ( q ) is either a highest or lowest weight module, or else a uniformly bounded module, where the parameter q is a nonzero complex number. We also classify quasifinite irreducible highest weight B ( q ) -modules and irreducible B ( q ) -modules of the intermediate series. In particular, we obtain that an irreducible B ( q ) -module of the intermediate series may be a nontrivial extension of a V ir -module of the intermediate series if q is half of a negative integer, where V ir is a subalgebra of B ( q ) isomorphic to the Virasoro algebra.

Factorization of the canonical bases for higher-level Fock spaces

AbstractThe level l Fock space admits canonical bases $\mathcal{G}_{e}$ and $\smash{\mathcal{G}_{\infty}}$. They correspond to $\smash{\mathcal{U}_{v}(\widehat{\mathfrak{sl}}_{e})}$ and $\mathcal{U}_{v}({\mathfrak{sl}}_{\infty})$-module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in ℕ[v]. Restriction to the highest-weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki–Koike algebras.

Representations of affine Nappi–Witten algebras

In this paper, we study the representation theory for the affine Lie algebra H ˆ 4 associated to the Nappi–Witten model H 4 . We classify all the irreducible highest weight modules of H ˆ 4 . Furthermore, we give a necessary and sufficient condition for each H ˆ 4 -(generalized) Verma module to be irreducible. For reducible ones, we characterize all the linearly independent singular vectors. Finally, we construct Wakimoto type modules for these Lie algebras and interpret this construction in terms of vertex operator algebras and their modules.

Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C2-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.

Representations of a symplectic type subalgebra of $W^N_{\infty }$W∞N

In this paper we classify the irreducible quasifinite highest weight modules over the symplectic type Lie subalgebra of the Lie algebra of all regular differential operators on circle that kill constants. We also realize them in terms of the representations theory of the complex Lie algebra \documentclass[12pt]{minimal}\begin{document}$g\ell _\infty ^{[m]}$\end{document}gℓ∞[m] of infinite matrices with a finite number of non-zero diagonals with entries in the algebra of truncated polynomials and the corresponding subalgebras of type C.

Combinatorics of character formulas for the lie superalgebra $ \mathfrak{g}\mathfrak{l}\left( {m,n} \right) $

Let \( \mathfrak{g} \) be the Lie superalgebra \( \mathfrak{g}\mathfrak{l}\left( {m,n} \right) \). Algorithms for computing the composition factors and multiplicities of Kac modules for \( \mathfrak{g} \) were given by the second author, [12] and by J. Brundan [1]. We give a combinatorial proof of the equivalence between the two algorithms. The proof uses weight and cap diagrams introduced by Brundan and C. Stroppel, and cancelations between paths in a graph \( \mathcal{G} \) defined using these diagrams. Each vertex of \( \mathcal{G} \) corresponds to a highest weight of a finite dimensional simple module, and each edge is weighted by a nonnegative integer. If \( \mathcal{E} \) is the subgraph of \( \mathcal{G} \) obtained by deleting all edges of positive weight, then \( \mathcal{E} \) is the graph that describes nonsplit extensions between simple highest weight modules. We also give a procedure for finding the composition factors of any Kac module, without cancelation. This procedure leads to a second proof of the main result.

Quasi-finite highest weight modules over WN∞

In this paper, we classify the irreducible quasi-finite highest weight modules of the Lie subalgebras WN∞, p of the Lie algebra of matrix differential operators on the circle. We also realize the WN∞-modules in terms of the representation theory of the complex Lie algebra gℓ[m]∞ of infinite matrices with a finite number of nonzero diagonals with entries in the algebra of truncated polynomials.

Modular invariant partition functions for noncompactG/Ad(H)models

We propose a spectrum for a class of gauged noncompact $G/Ad(H)$ Wess-Zumino-Novikov-Witten models, including spectrally flowed images of highest, lowest, and mixed extremal weight modules. These are combined into blocks whose characters, due to the Lorentzian signature of the target space, are divergent and treated as formal expressions in need of regularization. Assuming that this is possible, we show that these extended characters transform linearly under modular transformations, and can be used to write down modular invariant partition functions.

Generalized MICZ-Kepler problems and unitary highest weight modules

For each integer n ⩾ 1, we demonstrate that a (2n + 1)-dimensional generalized MICZ-Kepler problem has a Spin(2, 2n + 2) dynamical symmetry which extends the manifest Spin(2n + 1) symmetry. The Hilbert space of bound states is shown to form a unitary highest weight Spin(2, 2n + 2)-module with the minimal positive Gelfand–Kirillov dimension. As a byproduct, we obtain a simple geometric realization for such a unitary highest weight Spin(2, 2n + 2)-module.