Highest Weight Modules Research Articles

Exponents Associated with Y-Systems and their Relationship with q-Series

Let $X_r$ be a finite type Dynkin diagram, and $\ell$ be a positive integer greater than or equal to two. The $Y$-system of type $X_r$ with level $\ell$ is a system of algebraic relations, whose solutions have been proved to have periodicity. For any pair $(X_r, \ell)$, we define an integer sequence called exponents using formulation of the $Y$-system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type $X_r$, and prove this conjecture for $(A_1,\ell)$ and $(A_r, 2)$ cases. We point out that a specialization of this conjecture gives a relationship between the exponents and the asymptotic dimension of an integrable highest weight module of an affine Lie algebra. We also give a point of view from $q$-series identities for this relationship.

Integrable modules for twisted toroidal extended affine Lie algebras

In this paper we classify the irreducible integrable modules for the twisted toroidal extended affine Lie algebras (twisted toroidal EALA, in short) with finite dimensional weight spaces when the finite dimensional center acts non-trivially. Using an automorphism we reduce to the case where K0 acts non-trivially and Ki, 1≤i≤n act trivially. Twisted toroidal EALA has a natural triangular decomposition and we prove that any irreducible integrable module of it with finite dimensional weight spaces is a highest weight module with respect to the above triangular decomposition. The highest weight space is an irreducible module for the zeroth component of the twisted toroidal EALA. We then describe the highest weight space in detail.

Finite-Dimensional Representations of Hyper Multicurrent and Multiloop Algebras

We investigate the categories of finite-dimensional representations of multicurrent and multiloop hyperalgebras in positive characteristic, i.e., the hyperalgebras associated to the multicurrent algebras $\mathfrak {g}\otimes \mathbb {C}[t_{1},\ldots ,t_{n}]$ and to the multiloop algebras $\mathfrak {g}\otimes \mathbb {C}[t_{1}^{\pm 1},\ldots ,t_{n}^{\pm 1}]$ , where $\mathfrak {g}$ is any finite-dimensional complex simple Lie algebra. The main results are the construction of the universal finite-dimensional highest-weight modules and a classification of the irreducible modules in each category. In the characteristic zero setting we also provide a relationship between these modules.

Modules for loop Affine-Virasoro algebras

In this paper, we study the representations of loop Affine-Virasoro algebras. As they have canonical triangular decomposition, we define Verma modules and their irreducible quotients. We give necessary and sufficient condition for a irreducible highest weight module to have finite dimensional weight spaces. We prove that an irreducible integrable module is either a highest weight module or a lowest weight module whenever the canonical central element acts non-trivially. At the end, we construct Affine central operators for each integer and they commute with the action of the Affine Lie algebra.

Integrating over quiver variety and BPS/CFT correspondence

We show the vertex operator formalism for the quiver gauge theory partition function and the qq-character of the highest weight module on quiver, both associated with the integral over the quiver variety.

On Guay’s Evaluation Map for Affine Yangians

We give a detailed proof of the existence of evaluation map for affine Yangians of type A to clarify that it needs an assumption on parameters. This map was first found by Guay but a proof of its well-definedness and the assumption have not been written down in the literature. We also determine the highest weights of evaluation modules defined as the pull-back of integrable highest weight modules of the affine Lie algebra $\hat{\mathfrak{gl}}_N$ by the evaluation map.

Trigonometric Lie algebras, affine Lie algebras, and vertex algebras

In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra Aˆ of Lie algebra A=gl∞⊕gl∞ with respect to certain automorphism groups. We then prove that restricted modules of level ℓ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras VAˆ(ℓ,0) (or VAˆ(2ℓ,0)). Furthermore, we determine irreducible modules and equivariant quasi modules for the simple vertex algebra LAˆ(ℓ,0) with ℓ a positive integer. In particular, we prove that every quasi-finite unitary highest weight (irreducible) module of level ℓ for type A trigonometric Lie algebra gives rise to an irreducible equivariant quasi LAˆ(ℓ,0)-module.

Odd Singular Vector Formula for General Linear Lie Superalgebras

We establish a closed formula for a singular vector of weight $\lambda-\beta$ in the Verma module of highest weight $\lambda$ for Lie superalgebra $\mathfrak{gl}(m|n)$ when $\lambda$ is atypical with respect to an odd positive root $\beta$. It is further shown that this vector is unique up to a scalar multiple, and it descends to a singular vector, again unique up to a scalar multiple, in the corresponding Kac module when both $\lambda$ and $\lambda-\beta$ are dominant integral.

Simple restricted modules for Neveu-Schwarz algebra

In this paper, we give a construction of simple modules generalizing both highest weight and Whittaker modules for the Neveu-Schwarz algebra, in the spirit of the work of Mazorchuk and Zhao on simple Virasoro modules. We establish a 1-1 correspondence between simple restricted Neveu-Schwarz modules and simple modules of a family of finite dimensional solvable Lie superalgebras associated to the Neveu-Schwarz algebra. Moreover, for two of these superalgebras all simple modules are classified.

Classification of irreducible Harish-Chandra modules over gap-p Virasoro algebras

We prove that any irreducible Harish-Chandra module for a class of Lie algebras, which we call gap-p Virasoro algebras, must be a highest weight module, a lowest weight module, or a module of intermediate series. These algebras are closely related to the Heisenberg-Virasoro algebra and the algebra of derivations over a quantum torus. They also contain subalgebras which are isomorphic to the Virasoro algebra Vir, but graded by pZ (unlike Vir by Z).

A new class of irreducible Virasoro modules from tensor product

In this paper, we obtain a class of Virasoro modules by taking tensor products of the irreducible Virasoro modules Ω(λ,α,h) and Ω(μ,β) with irreducible highest weight modules V(θ,h) or with irreducible Virasoro modules Indθ(N) defined in [20]. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for two of them to be isomorphic. We also compare these modules with other known non-weight Virasoro modules, showing that most of these simple modules are new.

Harish-Chandra modules for divergence zero vector fields on a torus

Let $A=\mathbb{C}[t_1^{\pm1},t_2^{\pm1}]$ be the algebra of Laurent polynomials in two variables and $B$ be the set of skew derivations of $A$. Let $L$ be the universal central extension of the derived Lie subalgebra of the Lie algebra $A\rtimes B$. Set $\widetilde{L}=L\oplus\mathbb{C} d_1\oplus\mathbb{C} d_2$, where $d_1$, $d_2$ are two degree derivations. A Harish-Chandra module is defined as an irreducible weight module with finite dimensional weight spaces. In this paper, we prove that a Harish-Chandra module of the Lie algebra $\widetilde{L}$ is a uniformly bounded module or a generalized highest weight (GHW for short) module. Furthermore, we prove that the nonzero level Harish-Chandra modules of $\widetilde{L}$ are GHW modules. Finally, we classify all the GHW Harish-Chandra modules of $\widetilde{L}$.

Gelfand-Tsetlin representations of finite W-algebras

We construct explicitly a large family of new simple modules for an arbitrary finite W-algebra of type A. A basis of these modules is given by the Gelfand-Tsetlin tableaux whose entries satisfy certain sets of relations. Characterization and an effective method of constructing such admissible relations are given. In particular we describe a family of simple infinite dimensional highest weight relation modules. We also prove a sufficient condition for the simplicity of tensor product of two highest weight relation modules and establish the simplicity of the tensor product any number of relation modules with generic highest weights. This extends the results of Molev to infinite dimensional highest weight modules.

Polyhedral realizations of crystal bases B(λ) for quantum algebras of nonexceptional affine types

We give explicit forms of the crystal bases B(λ) for the integrable highest weight modules of the quantum affine algebras for A2n(2), A2n−1(2), Bn(1), Cn(1), and Dn+1(2).

Quantum Exceptional Group G2 and its Semisimple Conjugacy Classes

We construct quantization of semisimple conjugacy classes of the exceptional group G = G2 along with and by means of their representations on highest weight modules over the quantum group U_{q}(mathfrak {g}). With every point t of a fixed maximal torus we associate a highest weight module Mt over U_{q}(mathfrak {g}) and realize the quantized polynomial algebra of the class of t by linear operators on Mt. Quantizations corresponding to points of the same orbit of the Weyl group are isomorphic.

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.

Simple Singular Whittaker Modules Over the Schrödinger Algebra

There are no simple singular Whittaker modules over most of important algebras, such as simple complex finite-dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, the Heisenberg–Virasoro algebra and the Schrodinger–Witt algebra. In this paper, however, we construct simple singular Whittaker modules over the Schrodinger algebra. Moreover, simple singular Whittaker modules over the Schrodinger algebra are classified. As a result, simple modules for the Schrodinger algebra which are locally finite over the positive part are completely classified. We also give characterizations of simple highest weight modules and simple singular Whittaker modules.

Generalized Oscillator Representations of the Twisted Heisenberg-Virasoro Algebra

In this paper, we first obtain a general result on sufficient conditions for tensor product modules to be simple over an arbitrary Lie algebra. We classify simple modules with a nice property over the infinite-dimensional Heisenberg algebra ${\H}$, and then obtain a lot of simple modules over the twisted Heisenberg-Virasoro algebra $\LL$ from generalized oscillator representations of $\LL$ by extending these $\H$-modules. We give the necessary and sufficient conditions for Whittaker modules over $\LL$ (in the more general setting) to be simple. We use the "shifting technique" to determine the necessary and sufficient conditions for the tensor products of highest weight modules and modules of intermediate series over $\LL$ to be simple. At last we establish the "embedding trick" to obtain a lot more simple $\LL$-modules.

A Weyl Module Stratification of Integrable Representations

We construct a filtration on an integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each global Weyl module there is given by the corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as the graded dimension of the space of conformal coinvariants. In addition, as an application of the level one case of the main result, we realize global Weyl modules of current algebras of type $${\mathsf{ADE}}$$ in terms of Schubert subvarieties of thick affine Grassmanian, as predicted by Boris Feigin.

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.