Irreducible Weight Modules Research Articles

Dirac cohomology, branching laws and Wallach modules

The idea of using Dirac cohomology to study branching laws was initiated by Huang, et al. (2013) [11]. One of their results says that the Dirac cohomology of π completely determines π|K, where π is any irreducible unitarizable highest weight (g,K) module. This paper aims to develop this idea for the exceptional Lie groups E6(−14) and E7(−25): we recover the K-spectrum of the Wallach modules from their Dirac cohomology.

Structure and Representations for Electrical Lie Algebra of Type D5

In this paper, we prove that the electrical Lie algebra [Formula: see text] is isomorphic to the semidirect product of [Formula: see text] and a 2-step nilpotent Lie algebra. Furthermore, we classify the irreducible highest weight modules for [Formula: see text].

Defining Relations for Minimal Unitary Quantum Affine W-Algebras

We prove that any unitary highest weight module over a universal minimal quantum affine W-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine W-algebras and the list of all their irreducible positive energy modules. We also classify all irreducible highest weight modules for the simple affine vertex algebras in the cases when the associated simple minimal W-algebra is unitary.

Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

Let M be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra G, where G is gl(m|n), osp(2m|2n) or spo(2m|2n). We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for G on M can be obtained from the singular eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. As a consequence, the (super) Gaudin Hamiltonians for G are diagonalizable on the space spanned by singular vectors of M and hence on M. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra so(2n) and the symplectic Lie algebra sp(2n), on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules.

Descriptions of crystal [formula omitted] for E6 and E7 types via tableaux and Kashiwara embedding

Let Uq(g) be the quantized universal enveloping algebra for a Lie algebra g, and let Vq(λ) be the irreducible highest weight module for Uq(g). The crystal base B(λ) is a colored directed graph that captures the structure of Vq(λ) and the action of Uq(g) on Vq(λ) in a rudimentary manner. Likewise, the crystal base B(∞) holds the bare structure of the negative part Uq−(g).In this work, we describe realizations of the crystal B(λ) via two separate approaches for the cases when the base Lie algebra g is of E6 and E7 types. Our first description relies on the fact that B(λ) appears as a connected component within the much larger crystal B(∞)⊗{rλ}, where {rλ} is a certain single-element crystal. Choosing to represent elements of B(∞) with marginally large tableaux, we identify those elements belonging to the mentioned connected component. Our second description of B(λ) is a translation of the first description into one involving the Kashiwara embedding, which is an embedding of B(∞) into a tensor product of a series of much simpler crystals.

Tensor product decomposition of crystal bases of type A by K-hives

The crystal basis of an irreducible highest weight module of type A is realized by a combinatorial model called K-hives. In this paper, we give the tensor product decomposition map of crystal bases of irreducible highest weight modules of type A in terms of K-hives. In the process of constructing the decomposition map, we provide a combinatorial description on K-hives of the decomposition map.

Piecewise dominant sequences and the cocenter of the cyclotomic quiver Hecke algebras

In this paper we study the cocenter of the cyclotomic quiver Hecke algebra $${\mathscr {R}}^\Lambda _\alpha $$ associated to an arbitrary symmetrizable Cartan matrix $$A=(a_{ij})_{{i,j}\in I}$$ , $$\Lambda \in P^+$$ and $$\alpha \in Q_n^+$$ . We introduce a notion called “piecewise dominant sequence” and use it to construct some explicit homogeneous elements which span the cocenter of $${\mathscr {R}}^\Lambda _\alpha $$ . Our first main result shows that the minimal (resp., maximal) degree component of the cocenter of $${\mathscr {R}}^\Lambda _\alpha $$ is spanned by the image of some KLR idempotent $$e(\nu )$$ (resp., some monomials $$Z(\nu )e(\nu )$$ on KLR $$x_k$$ and $$e(\nu )$$ generators), where each $$\nu \in I^\alpha $$ is piecewise dominant. As an application, we show that any weight space $$L(\Lambda )_{\Lambda -\alpha }$$ of the irreducible highest weight module $$L(\Lambda )$$ over $${\mathfrak {g}}(A)$$ is nonzero (equivalently, $${\mathscr {R}}_{\alpha }^{\Lambda }\ne 0$$ ) if and only if there exists a piecewise dominant sequence $$\nu \in I^\alpha $$ . Finally, we show that the Indecomposability Conjecture on $${\mathscr {R}}^\Lambda _\alpha (K)$$ holds if it holds when K is replaced by a field of characteristic 0. In particular, this implies $${\mathscr {R}}^\Lambda _\alpha (K)$$ is indecomposable when K is a field of arbitrary characteristic and $${\mathfrak {g}}$$ is symmetric and of finite type.

Unitarity of Minimal W-Algebras and Their Representations I

We begin a systematic study of unitary representations of minimal W-algebras. In particular, we classify unitary minimal W-algebras and make substantial progress in classification of their unitary irreducible highest weight modules. We also compute the characters of these modules.

Generalized imaginary Verma and Wakimoto modules

We develop a general technique of constructing new irreducible weight modules for any affine Kac-Moody algebra using the parabolic induction, in the case when the Levi factor of a parabolic subalgebra is infinite-dimensional and the central charge is nonzero. Our approach unifies and generalizes all previously known results with imposed restrictions on inducing modules. We also define generalized imaginary Wakimoto modules which provide an explicit realization for generic generalized imaginary Verma modules.

Admissible-level $$\mathfrak {sl}_3$$ minimal models

The first part of this work uses the algorithm recently detailed in arXiv:1906.02935 to classify the irreducible weight modules of the minimal model vertex operator algebra $L_k(\mathfrak{sl}_3)$, when the level $k$ is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of $\hat{\mathfrak{sl}}_3$ on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family's parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in arXiv:2003.10148, these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level $\mathfrak{sl}_3$ minimal models. The second part of this work applies the standard module formalism to compute these explicitly when $k=-\frac32$. We expect that the methodology developed here will apply in much greater generality.

Irreducible modules of toroidal Lie algebras arising from ϕϵ-coordinated modules of vertex algebras

In this paper, for every ϵ∈Z, we introduce an extension of the 2-toroidal Lie algebra by certain derivations. Based on the ϕϵ-coordinated modules theory for vertex algebras, we give an explicit realization of a class of irreducible highest weight modules for this extended toroidal Lie algebra. When ϵ=1, this affords a realization of certain irreducible modules for the toroidal extended affine Lie algebras first constructed by Billig.

Representations of twisted affine Lie superalgebras–A recognition theorem

Over the past three decades, representation theory of affine Lie (super)algebras has been a research spotlight in both areas Mathematics and Physics. The even parts of almost all affine Lie superalgebras L contain two affine Lie algebras; say L0(1) and L0(2). Irreducible finite weight modules (f.w.m's for short) are the first interesting class of modules over such an L. The structure of these modules strongly depends on whether of root vectors corresponding to real roots of L0(1) and L0(2) act locally nilpotently or not. We know that there is no nonzero irreducible f.w.m. of nonzero level over L for which root vectors corresponding to nonzero real roots of both L0(1) and L0(2) act locally nilpotently. This leads us to define quasi-integrable modules and study the case that root vectors corresponding to nonzero real roots of one of L0(1) and L0(2) act locally nilpotently. We give a recognition theorem for quasi-integrable modules over twisted affine Lie superalgebras.

Crystal structure on K-hives of type A

We show that a set of K-hives is isomorphic as a crystal to the crystal basis of the irreducible highest weight module on a quantized enveloping algebra of type A. Also, we explicitly describe the crystal structure on K-hives. To introduce this structure, we construct an embedding similar to the Far-Eastern reading for K-hives. Moreover, we provide an explicit description of the crystal structure by constructing the Middle-Eastern reading for K-hives.

A new class of irreducible modules over the affine-Virasoro algebra of type A1

In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type A1 by taking tensor products of a finite number of irreducible modules M(λ,α,β,γ) with irreducible highest weight modules V(η,ϵ,θ). We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for such two modules to be isomorphic. We also compare these modules with other known non-weight modules, showing that these irreducible modules are new.

Global bases for quantum Borcherds–Bozec algebras

We provide an explicit construction of global bases for quantum Borcherds–Bozec algebras and their irreducible highest weight modules. Moreover, we give a new presentation for quantum Borcherds–Bozec algebras in terms of primitive generators.

Tensor product weight modules for the mirror Heisenberg-Virasoro algebra

In this paper, we study irreducible weight modules with infinite dimensional weight spaces over the mirror Heisenberg-Virasoro algebra D . More precisely, the necessary and sufficient conditions for the tensor products of irreducible highest weight modules and irreducible modules of intermediate series over D to be irreducible are determined by using “shifting technique”. This leads to a family of new irreducible weight modules over D . Then we obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight modules and modules of intermediate series are isomorphic respectively. Also we discuss submodules of the tensor product module when it is not irreducible.

Multi-grounded partitions and character formulas

We introduce a new generalisation of partitions, multi-grounded partitions, related to ground state paths indexed by dominant weights of Lie algebras. We use these to express characters of irreducible highest weight modules of Kac–Moody algebras of affine type as generating functions for multi-grounded partitions. This generalises the approach of our previous paper, where only irreducible highest weight modules with constant ground state paths were considered, to all ground state paths. As an application, we compute the characters of the level 1 modules of the affine Lie algebras A2n(2)(n≥2), Dn+1(2)(n≥2), A2n−1(2)(n≥3), Bn(1)(n≥3), and Dn(1)(n≥4).

Relaxed highest-weight modules III: Character formulae

This is the third of a series of articles devoted to the study of relaxed highest weight modules over vertex operator algebras. Relaxed highest weight modules over affine vertex algebras associated to higher rank Lie algebras Aℓ and arbitrary non-critical levels whose top spaces are cuspidal representations of Aℓ are extensively studied. In particular, the string functions of irreducible relaxed highest weight modules whose top spaces are cuspidal Aℓ-modules are shown to be the quotients by a power of the Dedekind eta series of the q-characters of irreducible ordinary modules over affine W-algebras associated with the minimal nilpotent elements of Aℓ.

Irreducible tensor product modules over the affine-Virasoro algebra of type A1

In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type A 1 by taking the tensor product of irreducible U ( h ) -free modules of rank 1 with irreducible highest weight modules. The irreducibility and the isomorphism classes of these modules are determined. Moreover, we show that these tensor product modules are different from the known non-weight modules. Finally, we realize some tensor product modules as induced modules from modules over certain subalgebras of the affine-Virasoro algebra of type A 1 , and give sufficient and necessary conditions for these induced modules to be reducible.

Representations of the q-Klein-bottle Lie algebra

We give the necessary and sufficient conditions for all weight spaces of the highest weight irreducible module V(ϕ) of the q-Klein-bottle Lie algebra B to be finite dimensional. We prove that the irreducible highest weight module V(ϕ) equals to the Verma B-module V¯(ϕ) if and only if V(ϕ) has at least one infinite-dimensional weight space. We also give the maximal proper submodule of the Verma module V¯(ϕ) and the e-character of the irreducible highest weight B-module V(ϕ) when the highest weight ϕ satisfies some natural conditions. Furthermore, we give the classification of the Harish-Chandra B-modules with nontrivial central charge.