Irreducible Weight Research Articles

On the First Order Cohomology of Infinite-Dimensional Unitary Groups

We determine precisely for which irreducible unitary highest weight representation of the group U(∞), the countable direct limit of the finite-dimensional unitary groups U(n), the corresponding 1-cohomology space H 1 does not vanish. This occurs in particular if a highest weight, viewed as an integer-valued function on ℕ, is finitely supported. In a second step, we extend the finitely supported highest weight representations to norm-continuous unitary representations of the Banach-completions U p (ℓ 2 ) of the direct limit U(∞) with respect to the pth Schatten norm for 1≤p≤∞. For p<∞, the corresponding 1-cohomology spaces H 1 do not vanish either, except in three cases. We conclude that these groups do not have Kazhdan’s Property (T). On the other hand, for p=∞, the first cohomology spaces all vanish because U ∞ (ℓ 2 ) has property (FH) as a bounded topological group.

A lower bound for the dimension of a highest weight module

For each integer t &gt; 0 t&gt;0 and each simple Lie algebra g \mathfrak {g} , we determine the least dimension of an irreducible highest weight representation of g \mathfrak {g} whose highest weight has width t t . As a consequence, we classify all irreducible modules whose dimension equals a product of two primes. This consequence, which was in fact the driving force behind our paper, answers a question of N. Katz.

QHWM of the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo differential operators

In this paper we classify the irreducible quasifinite highest weight modules over the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo-differential operators. We also realize them in terms of the irreducible quasifinite highest weight modules of the Lie algebras of infinite matrices with finitely many nonzero diagonals and its classical Lie subalgebras of types B, C and D.

Irreducible Weight Modules with a Finite-Dimensional Weight Space over the Twisted N = 1 Schrödinger-Neveu-Schwarz Algebra

It is shown that there are no simple mixed modules over the twisted N = 1 Schrödinger-Neveu-Schwarz algebra, which implies that every irreducible weight module over it with a non-trivial finite-dimensional weight space is a Harish-Chandra module.

Rigged configuration descriptions of the crystals B(∞) and B(λ) for special linear Lie algebras

The rigged configuration realization $RC(\infty)$ of the crystal $B(\infty)$ was originally presented as a certain connected component within a larger crystal. In this work, we make the realization more concrete by identifying the elements of $RC(\infty)$ explicitly for the $A_n$-type case. Two separate descriptions of $RC(\infty)$ are obtained. These lead naturally to isomorphisms $RC(\infty)\cong T(\infty)$ and $RC(\infty)\cong \bar{T}(\infty)$, i.e., those with the marginally large tableau and marginally large reverse tableau realizations of $B(\infty)$, that may be computed explicitly. We also present two descriptions of the irreducible highest weight crystal $B(\lambda)$ in terms of rigged configurations. These are obtained by combining our two descriptions of $RC(\infty)$, the two mentioned isomorphisms, and two existing realizations of $B(\lambda)$ that were based on $T(\infty)$ and $\bar{T}(\infty)$.

Parabolic degeneration of rational Cherednik algebras

We introduce parabolic degenerations of rational Cherednik algebras of complex reflection groups, and use them to give necessary conditions for finite-dimensionality of an irreducible lowest weight module for the rational Cherednik algebra of a complex reflection group, and for the existence of a non-zero map between two standard modules. The latter condition reproduces and enhances, in the case of the symmetric group, the combinatorics of cores and dominance order, and in general shows that the c-ordering on category $$\mathcal {O}_c$$ may be replaced by a much coarser ordering. The former gives a new proof of the classification of finite dimensional irreducible modules for the Cherednik algebra of the symmetric group.

A note on the zeroth products of Frenkel–Jing operators

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

Rigged Configurations for all Symmetrizable Types

In an earlier work, the authors developed a rigged configuration model for the crystal $B(\infty)$ (which also descends to a model for irreducible highest weight crystals via a cutting procedure). However, the result obtained was only valid in finite types, affine types, and simply-laced indefinite types. In this paper, we show that the rigged configuration model proposed does indeed hold for all symmetrizable types. As an application, we give an easy combinatorial condition that gives a Littlewood-Richardson rule using rigged configurations which is valid in all symmetrizable Kac-Moody types.

On Free Field Realizations of W(2,2)-Modules

The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra $\mathcal H$ at level zero as modules for the $W(2,2)$-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight ${\mathcal H}$-module is irreducible as $W(2,2)$-module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly $W(2,2)$ vertex algebra.

Reducibility of matrix weights

In this paper we discuss the notion of reducibility for matrix weights and introduce a real vector space $\mathcal C_\mathbb{R}$ which encodes all information about the reducibility of $W$. In particular a weight $W$ reduces if and only if there is a non-scalar matrix $T$ such that $TW=WT^*$. Also, we prove that reducibility can be studied by looking at the commutant of the monic orthogonal polynomials or by looking at the coefficients of the corresponding three term recursion relation. A matrix weight may not be expressible as direct sum of irreducible weights, but it is always equivalent to a direct sum of irreducible weights. We also establish that the decompositions of two equivalent weights as sums of irreducible weights have the same number of terms and that, up to a permutation, they are equivalent. We consider the algebra of right-hand-side matrix differential operators $\mathcal D(W)$ of a reducible weight $W$, giving its general structure. Finally, we make a change of emphasis by considering reducibility of polynomials, instead of reducibility of matrix weights.

Irreducible weight modules for the Virasoro-like algebra and its q-analog

ABSTRACTIn this paper, using Larsson’s functor with irreducible 𝔰𝔩2-modules V, we construct a class of ℤ2-graded modules for the Virasoro-like algebra and its q-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.

The combinatorial category of Andersen, Jantzen and Soergel and filtered moment graph sheaves

We give an overview on the series of articles (Fiebig and Lanini, Filtered moment graph sheaves, arXiv:1508.05579 , 2015, Fiebig and Lanini, Periodic structures on affine moment graphs I: dualities and translation functors, arXiv:1504.01699 , 2015, Fiebig and Lanini, Periodic structures on affine moment graphs II: multiplicities and modular representations (in preparation)) that aims at introducing a new approach towards the “combinatorial” category introduced by Andersen, Jantzen and Soergel in their work on Lusztig’s conjecture on the irreducible highest weight characters of modular algebraic groups.

Partial classification of modules for the algebra of skew-derivations over the d-dimensional torus

For the algebra of skew-derivations over a torus, we classify the irreducible weight modules with some restrictions.

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.

A new functor from D 5-Mod to E 6-Mod

We find a new representation of the simple Lie algebra of type E 6 on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen’s idea of mixed product, we construct a new functor from D 5-Mod to E 6-Mod. A condition for the functor to map a finite-dimensional irreducible D 5-module to an infinite-dimensional irreducible E 6-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant’s characteristic identities plays a key role.

Actions of the quantum toroidal algebra of type sl2 on the space of vertex operators for Uq(gl2̂) modules

Highest weight modules for Uq(gl2̂) are endowed with a structure of modules for the quantum toriodal algebra Uκ of type sl2. Using this, we define Uκ actions on the space of vertex operators for irreducible highest weight Uq(gl2̂) modules. Highest or lowest weight vectors of the thus obtained Uκ modules are expressed in terms of an intertwiner for Uq(sl2̂) modules and an extra boson. The submodules generated by these vectors are investigated.

Classification of irreducible integrable highest weight modules for current Kac–Moody algebras

This paper classifies irreducible, integrable highest weight modules for “current Kac–Moody Algebras” with finite-dimensional weight spaces. We prove that these modules turn out to be modules of appropriate direct sums of finitely many copies of Kac–Moody Lie algebras.

The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

In this paper we study in detail algebraic properties of the algebra $\mathcal D(W)$ of differential operators associated to a matrix weight of Gegenbauer type. We prove that two second order operators generate the algebra, indeed $\mathcal D(W)$ is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra $\mathcal D(W)$ is a finitely-generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra $\mathcal D(W)$ and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra $\mathcal D(W)$ as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this paper represents a good step in this direction.

Generalized polynomial modules over the Virasoro algebra

Let $\mathcal{B}_r$ be the $(r+1)$-dimensional quotient Lie algebra of the positive part of the Virasoro algebra $\mathcal{V}$. Irreducible $\mathcal{B}_r$-modules were used to construct irreducible Whittaker modules in [MZ2] and irreducible weight modules with infinite dimensional weight spaces over $\mathcal{V}$ in [LLZ].In the present paper, we construct non-weight Virasoro modules $F(M, \Omega(\lambda,\beta))$ from irreducible $\mathcal{B}_r$-modules $M$ and $(\mathcal{A},\mathcal{V})$-modules $\Omega(\lambda,\beta)$. We give necessary and sufficient conditions for the Virasoro module $F(M, \Omega(\lambda,\beta))$ to be irreducible. Using the weighting functor introduced by J. Nilsson, we also we also give the isomorphism criterion for two $F(M, \Omega(\lambda,\beta))$.

Representations of D q (k[x

The algebra of quantum differential operators on graded algebras was introduced by V. Lunts and A. Rosenberg. D. Jordan, T. McCune and the second author have identified this algebra of quantum differential operators on the polynomial algebra with coefficients in an algebraically closed field of characteristic zero. It contains the first Weyl algebra and the quantum Weyl algebra as its subalgebras. In this paper we classify irreducible weight modules over the algebra of quantum differential operators on the polynomial algebra. Some classes of indecomposable modules are constructed in the case of positive characteristic and q root of unity.