Affine Kac-Moody Algebras Research Articles

Limits of Multiplicities in Excellent Filtrations and Tensor Product Decompositions for Affine Kac-Moody Algebras

We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for $\widehat{\mathfrak{sl}}_2$ in terms of partitions with bounded parts, which subsequently lead to certain partition identities.

Weight Representations of Admissible Affine Vertex Algebras

For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra $\widehat{\mathfrak{g}}$ induced from the following $\mathfrak{g}$-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from $\mathfrak{sl}_2$; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from $\mathfrak{sl}_3$; 3) all simple Gelfand-Tsetlin modules over $\mathfrak{sl}_3$. This in particular gives the classification of all simple positive energy weight representations of $V_k(\mathfrak{g})$ with finite dimensional weight spaces for $\mathfrak{g}=\mathfrak{sl}_3$.

Structure of parabolically induced modules for affine Kac–Moody algebras

The main result of the paper establishes the irreducibility of a large family of nonzero central charge induced modules over Affine Lie algebras for any non standard parabolic subalgebra. It generalizes all previously known partial results and provides a construction of many new irreducible modules.

A remark on boundary level admissible representations

We point out that it is immediate by our character formula that in the case of a boundary level the characters of admissible representations of affine Kac–Moody algebras and the corresponding W-algebras decompose in products in terms of the Jacobi form ϑ11(τ,z).

A combinatorial formula expressing periodic R-polynomials

In 1980, Lusztig introduced the periodic Kazhdan–Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those of an affine Kac–Moody algebra at the critical level. The periodic Kazhdan–Lusztig polynomials can be computed by using another family of polynomials, called the periodic R-polynomials. In this paper, we prove a (closed) combinatorial formula expressing periodic R-polynomials in terms of the “doubled” Bruhat graph associated to a finite Weyl group and a finite root system.

Tau functions and Virasoro symmetries for Drinfeld–Sokolov hierarchies

For each Drinfeld–Sokolov integrable hierarchy associated to affine Kac–Moody algebra, we obtain a uniform construction of tau function by using tau-symmetric Hamiltonian densities, moreover, we represent its Virasoro symmetries as linear/nonlinear actions on the tau function. The relations between the tau function constructed in this paper and those defined for particular cases of Drinfeld–Sokolov hierarchies in the literature are clarified. We also show that, whenever the affine Kac–Moody algebra is simply-laced or twisted, the tau function of the Drinfeld–Sokolov hierarchy coincides with the solution of the corresponding Kac–Wakimoto hierarchy constructed from the principal vertex operator realization of the affine algebra.

Relaxed Highest Weight Modules from D-Modules on the Kashiwara Flag Scheme

The relaxed highest weight representations introduced by Feigin et al. are a special class of representations of the affine Kac-Moody algebra $\hat{\mathfrak{sl}_2}$, which do not have a highest (or lowest) weight. We formulate a generalization of this notion for an arbitrary affine Kac-Moody algebra $\mathfrak{g}$. We then realize induced $\mathfrak{g}$-modules of this type and their duals as global sections of twisted $\mathcal{D}$-modules on the Kashiwara flag scheme $X$ associated to $\mathfrak{g}$. The $\mathcal{D}$-modules that appear in our construction are direct images from subschemes of $X$ that are intersections of finite dimensional Schubert cells with their translate by a simple reflection. Besides the twist $\lambda$, they depend on a complex number describing the monodromy of the local systems we construct on these intersections. We describe the global sections of the $*$-direct images as a module over the Cartan subalgebra of $\mathfrak{g}$ and show that the higher cohomology vanishes. We obtain a complete description of the cohomology groups of the direct images as $\mathfrak{g}$-modules in the following two cases. First, we address the case when the intersection is isomorphic to $\mathbb{C}^{\times}$. Secondly, we address the case of the $*$-direct image from an arbitrary intersection when the twist is regular antidominant and the monodromy is trivial. For the proof of this case we introduce an exact auto-equivalence of the category of $\mathcal{D}$-modules $\text{Hol}(\lambda)$ induced by the automorphism of $X$ defined by a lift of a simple reflection. These results describe for the first time explicit non-highest weight $\mathfrak{g}$-modules as global sections on the Kashiwara flag scheme and extend several results of Kashiwara-Tanisaki to the case of relaxed highest weight representations.

JOSEPH IDEALS AND LISSE MINIMAL -ALGEBRAS

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_{2}$-cofinite)$W$-algebras that are not coming from admissible representations of affine Kac–Moody algebras.

AGT relations for abelian quiver gauge theories on ALE spaces

We construct level one dominant representations of the affine Kac–Moody algebra g l ̂ k on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution X k of the A k − 1 toric singularity C 2 / Z k . We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for g l ̂ k , which proves the AGT correspondence for pure N = 2 U ( 1 ) gauge theory on X k . We consider Carlsson–Okounkov type Ext-bundles over products of the moduli spaces and use their Euler classes to define vertex operators. Under the decomposition g l ̂ k ≃ h ⊕ s l ̂ k , these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra h and primary fields of s l ̂ k . We use these operators to prove the AGT correspondence for N = 2 superconformal abelian quiver gauge theories on X k .

Rationality of admissible affine vertex algebras in the category O

We study the vertex algebras associated with modular invariant representations of affine Kac-Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph's characteristic varieties. We show that an irreducible highest weight representation of a non-twisted affine Kac-Moody algebra at an admissible level k is a module over the associated simple affine vertex algebra if and only if it is an admissible representation whose integral root system is isomorphic to that of the vertex algebra itself. This in particular proves the conjecture of Adamovic and Milas on the rationality of admissible affine vertex algebras in the category O.

Lie algebra extensions of current algebras on S3

An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory.

Quantum dimensions and fusion rules for parafermion vertex operator algebras

The quantum dimensions and the fusion rules for the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A_1^{(1)} of level k are determined.

Automorphism group of parafermion vertex operator algebras

The automorphism group of parafermion vertex operator algebra associated with the irreducible highest weight module for the affine Kac–Moody algebra A1(1) was easily determined since the Virasoro primary vector of weight 3 in this parafermion vertex operator algebra is unique up to a scalar. However, it is highly nontrivial to determine the automorphism group of parafermion vertex operator algebra associated with the irreducible highest weight module for the affine Kac–Moody algebra An(1) with the rank n≥2. As the first step, in this paper, we determine the full automorphism group of parafermion vertex operator algebra associated with the irreducible highest weight module for the affine Kac–Moody algebra A2(1), which shows the idea for a complete determination for the full automorphism group of the parafermion vertex operator algebra associated with the irreducible highest weight module for any affine Kac–Moody algebra.

Soliton equations related to the affine Kac-Moody algebra D 4 (1)

We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D (1)4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.

Symmetric quiver Hecke algebras and R -matrices of quantum affine algebras, II

Let $\g$ be an untwisted affine Kac-Moody algebra of type $A^{(1)}_n$ $(n \ge 1)$ or $D^{(1)}_n$ $(n \ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$, Hernandez and Leclerc (\cite{HL11}) introduced a tensor subcategory $\CC_Q$ of the category of finite-dimensional integrable $\uqpg$-modules and proved that the Grothendieck ring of $\CC_Q$ is isomorphic to $\C [N]$, the coordinate ring of the unipotent group $N$ associated with $\g_0$. We apply the generalized quantum affine Schur-Weyl duality introduced in \cite{KKK13} to construct an exact functor $\F$ from the category of finite-dimensional graded $R$-modules to the category $\CC_Q$, where $R$ denotes the symmetric quiver Hecke algebra associated to $\g_0$. We prove that the homomorphism induced by the functor $\F$ coincides with the homomorphism of Hernandez and Leclerc and show that the functor $\F$ sends the simple modules to the simple modules.

Orthogonal symmetric affine Kac-Moody algebras

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification.

Localization of Free Field Realizations of Affine Lie Algebras

We use localization technique to construct new families of irreducible modules of affine Kac-Moody algebras. In particular, localization is applied to the first free field realization of the affine Lie algebra A_1^{(1)} or, equivalently, to imaginary Verma modules.

Distributions Defined by $q$-Supernomials, Fusion Products, and Demazure Modules

We prove asymptotic normality of the distributions defined by $q$-supernomials, which implies asymptotic normality of the distributions given by the central string functions and the basic specialization of fusion modules of the current algebra of $\frak{sl}_2$. The limit is taken over linearly scaled fusion powers of a fixed collection of irreducible representations. This includes as special instances all Demazure modules of the affine Kac-Moody algebra associated to $\frak{sl}_2$. Along with an available complementary result on the asymptotic normality of the basic specialization of graded tensors of the type $A$ standard representation, our result is a central limit theorem for a serious class of graded tensors. It therefore serves as an indication towards universal behavior: The central string functions and the basic specialization of fusion and, in particular, Demazure modules behave asymptotically normal, as the number of fusions scale linearly in an asymptotic parameter, $N$ say.

On the stable moment graph of an affine Kac–Moody algebra

In 1980 Lusztig proved a stabilisation property of the affine Kazhdan-Lusztig polynomials. In this paper we give a categorical version of such a result using the theory of sheaves on moment graphs. This leads us to associate with any Kac-Moody algebra its stable moment graph.

Some remarks on representations of Yang-Mills algebras

In this article, we present some new properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine Kac-Moody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043–1080 (2011)], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.