Irreducible Weight Modules Research Articles

CLASSIFICATION OF IRREDUCIBLE WEIGHT MODULES OVER THE TWISTED HEISENBERG–VIRASORO ALGEBRA

In this paper, all irreducible weight modules with finite dimensional weight spaces over the twisted Heisenberg–Virasoro algebra are determined. There are two different classes of them. One class is formed by simple modules of intermediate series, whose nonzero weight spaces are all 1-dimensional; the other class consists of the irreducible highest weight modules and lowest weight modules.

Verma Modules over Quantum Torus Lie Algebras

AbstractRepresentations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras . The center of now is generally infinite dimensional.In this paper, Z-graded Verma modules over and their corresponding irreducible highest weight modules are defined for some linear functions . Necessary and sufficient conditions for to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules e to be irreducible are obtained.

Monomial Realization of the Crystals Over U q (A ∞) and the Quantum Monster Algebra

We give new realizations of the crystal bases of the Verma modules and the irreducible highest weight modules over the quantum generalized Kac–Moody algebra U q (A ∞) and the quantum Monster algebra using Nakajima monomials. Moreover, another realization of the crystals B(∞) and B(λ) over U q (A ∞) using triangular matrices and tableaux are given.

Representations of four-derivation Lie algebras of Block type

Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on “polynomials” for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained.

Highest weight modules over pre-exp-polynomial Lie algebras

In this paper, we introduce pre-exp-polynomial Lie algebras which include loop algebras, Virasoro-like algebras and some quantum torus Lie algebras. We study “highest weight” representations of these Z n + 1 -graded Lie algebras. More precisely, we show that non-graded and graded irreducible highest weight modules with the same highest weight simultaneously have all finite-dimensional weight spaces or not, and they have all finite-dimensional weight spaces if and only if the highest weight is an exp-polynomial “highest weight”. We also show that non-graded and graded highest weight Verma modules with the same highest weight are simultaneously irreducible or not, and we determine necessary and sufficient conditions for a Verma module to be irreducible.

The structure of parafermion vertex operator algebras

It is proved that the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac–Moody algebra A 1 ( 1 ) of level k coincides with a certain W-algebra. In particular, a set of generators for the parafermion vertex operator algebra is determined.

Irreducible weight modules over the twisted Schrödinger-Virasoro algebra

It is shown that the support of an irreducible weight module over the Schrodinger-Virasoro Lie algebra with an infinite-dimensional weight space coincides with the weight lattice, and all nontrivial weight spaces of such a module are infinite-dimensional. As a by-product, it is obtained that every simple weight module over Lie algebra of this type with a nontrivial finite-dimensional weight space is a Harish-Chandra module.

Generalized Verma Modules over Lie Algebras of Weyl Type

For a field 𝔽 of characteristic 0 and an additive subgroup Γ of 𝔽, there corresponds a Lie algebra [Formula: see text] of generalized Weyl type. Given a total order of Γ and a weight Λ, a generalized Verma [Formula: see text]-module M(Λ, ≺) is defined. In this paper, the irreducibility of M(Λ, ≺) is completely determined. It is also proved that an irreducible highest weight module over the [Formula: see text]-infinity algebra [Formula: see text] is quasifinite if and only if it is a proper quotient of a Verma module.

Classification of irreducible weight modules over W-algebra W(2,2)

In this paper, we show that the support of an irreducible weight module over the W-algebra W(2,2), which has an infinite dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the W-algebra W(2,2), having a nontrivial finite dimensional weight space, is a Harish–Chandra module (and hence is either an irreducible highest or a lowest weight module or an irreducible module of the intermediate series).

Quantization of Lie Bialgebras, Part VI: Quantization of Generalized Kac–Moody Algebras

This paper is a continuation of the series of papers “Quantization of Lie bialgebras (QLB) I-V”. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with the tensor category structure defined by a Drinfeld associator. This equivalence is a generalization of the functor constructed previously by G. Lusztig and the second author. In particular, we answer positively a question of Drinfeld whether the characters of irreducible highest weight modules for quantized Kac-Moody algebras are the same as in the classical case. Moreover, our results are valid for the Lie algebra \(\mathfrak{g}(A)\) corresponding to any symmetrizable matrix A (not necessarily with integer entries), which answers another question of Drinfeld. We also prove the Drinfeld-Kohno theorem for the algebra \(\mathfrak{g}(A)\) (it was previously proved by Varchenko using integral formulas for solutions of the KZ equations).

Representations of the Schrödinger–Virasoro algebras

In this paper, it is proved that an irreducible weight module with finite-dimensional weight spaces over the Schrödinger–Virasoro algebras is a highest∕lowest weight module or a uniformly bounded module. Furthermore, indecomposable modules of the intermediate series over these algebras are completely determined.

Weighted projective spaces and minimal nilpotent orbits

We investigate (twisted) rings of differential operators on the resolution of singularities of an irreducible component X ¯ \overline X of O ¯ m i n ∩ n + \overline O_{\mathrm {min}}\cap \mathfrak n_+ (where O ¯ m i n \overline O_{\mathrm {min}} is the (Zariski) closure of the minimal nilpotent orbit of s p 2 n \mathfrak {sp}_{2n} and n + \mathfrak n_+ is the Borel subalgebra of s p 2 n \mathfrak {sp}_{2n} ) using toric geometry, and show that they are homomorphic images of a certain family of associative subalgebras of U ( s p 2 n ) U(\mathfrak {sp}_{2n}) , which contains the maximal parabolic subalgebra p \mathfrak p determining O ¯ min \overline O_{\min } . Further, using Fourier transforms on Weyl algebras, we show that (twisted) rings of well-suited weighted projective spaces are obtained from the same family of subalgebras. Finally, we investigate this family of subalgebras from the representation-theoretical point of view and, among other things, rediscover in a different framework irreducible highest weight modules for the UEA of s p 2 n \mathfrak {sp}_{2n} .

Signatures of Invariant Hermitian Forms on Irreducible Highest-Weight Modules

Perhaps the most important problem in representation theory in the 1970s and early 1980s was the determination of the multiplicity of com- position factors in a Verma module. This problem was settled by the proof of the Kazhdan-Lusztig Conjecture which states that the multiplicities may be computed via Kazhdan-Lusztig polynomials. In this paper, we introduce signed Kazhdan-Lusztig polynomials, a variation of Kazhdan-Lusztig polyno- mials which encodes signature information in addition to composition factor multiplicities and Jantzen filtration level. Careful consideration of Gabber and Joseph's proof of Kazhdan and Lusztig's recursive formula for computing Kazhdan-Lusztig polynomials and an application of Jantzen's determinant for- mula lead to a recursive formula for the signed Kazhdan-Lusztig polynomials. We use these polynomials to compute the signature of an invariant Hermitian form on an irreducible highest weight module. Such a formula has applications to unitarity testing. 1.1. The Unitary Dual Problem. In the 1930s, I.M. Gelfand introduced a broad programme in abstract harmonic analysis which would permit the transfer of dif- ficult problems in areas as distinct from analysis as topology to more tractable problems in algebra. Fourier analysis is just one incarnation of this programme. An unresolved component in Gelfand's programme is the classification of the irre- ducible unitary representations of a group, known as the unitary dual problem. In the case of a real reductive Lie group, the problem is equivalent to identifying all irreducible Harish-Chandra modules which admit a positive definite invariant Hermitian form. As Harish-Chandra modules may be constructed via an algebraic method introduced by Zuckerman in 1978 known as cohomological induction, it is of interest to study signatures of invariant Hermitian forms on cohomologically induced modules and to understand how positivity can fail. Cohomological induction is a two-step process in which we compose an induction functor with a Zuckerman functor i . The intermediate module in cohomological induction is a generalized Verma module which admits an invariant Hermitian form if the module to which induction was applied admits an invariant Hermitian form. Formulas for the signatures of invariant Hermitian forms on these intermediate modules may be used to compute signatures of forms on corresponding cohomo- logically induced modules (eg. (Wal84)). This motivates the study of invariant Hermitian forms on Verma modules (see (Wal84), (Yee05)) and irreducible highest weight modules.

Orthosymplectic Lie Superalgebras in Superspace Analogues of Quantum Kepler Problems

A Schrodinger type equation on the superspace \(\mathbb {R}^{D|2n}\) is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant “distance” away from the origin. An osp(2, D + 1|2n) dynamical supersymmetry for the system is explicitly constructed, and the bound states of the system are shown to form an irreducible highest weight module for this superalgebra. A thorough understanding of the structure of the irreducible module is obtained. This in particular enables the determination of the energy eigenvalues and the corresponding eigenspaces as well as their respective dimensions.

A combinatorial model for crystals of Kac-Moody algebras

We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be viewed as a discrete counterpart to the Littelmann path model. We describe crystal graphs and give a Littlewood-Richardson rule for decomposing tensor products of irreducible representations. The new model is based on the notion of a $\lambda$-chain, which is a chain of positive roots defined by certain interlacing conditions.

Quasifinite representations of the Lie superalgebra of quantum pseudodifferential operators

In this paper we extend general results obtained by Kac and Liberati [Lett. Math. Phys. 53, 11 (2000)] for quasifinite highest weight representations of Z-graded Lie algebras to 12Z-graded Lie superalgebras, and we apply these to classify the irreducible quasifinite highest weight modules of the Lie superalgebra of quantum pseudodifferential operators.

A (q,γ) analog of the W1+∞ algebra

A (q,γ) analog of the W1+∞ algebra is introduced. Irreducible quasifinite highest weight modules of this algebra and R matrices acting on a tensor product of these modules are investigated. A connection with the q deformed Virasoro and WN algebras is also discussed.

Visible actions on symmetric spaces

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $\sigma$ such that $\sigma |_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric space $D = G/K$ the action of any symmetric subgroup $H$ is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold $S$. Our geometric results provide a uniform proof of various multiplicity-free theorems of irreducible highest weight modules when restricted to reductive symmetric pairs, for both classical and exceptional cases, for both finite and infinite dimensional cases, and for both discrete and continuous spectra.

Lowest weight modules of [formula omitted] of minimal Gelfand–Kirillov dimension

This paper is to classify genuine irreducible lowest weight modules of Sp 2 n ( R ) ˜ , of Gelfand–Kirillov dimension n.

Irreducible representations for the abelian extension of the Lie algebra of diffeomorphisms of tori in dimensions greater than 1

We classify the irreducible weight modules of the abelian extension of the Lie algebra of diffeomorphisms of tori of dimension greater than 1, with finite-dimensional weight spaces.