Given a finite-dimensional complex simple Lie algebra $\mathfrak{g}$ and a complex number $κ$, let $V^κ(\mathfrak{g})$ be the associated universal affine vertex algebra. Gorelik and Kac [GK07] gave a sufficient and necessary condition for $V^κ(\mathfrak{g})$ to be simple. In this paper, for simply-laced $\mathfrak{g}$ and non-critical $κ$, we determine the weights of singular vectors of $V^κ(\mathfrak{g})$ with minimal conformal weights, when $V^κ(\mathfrak{g})$ is not simple. We further determine all the longest Weyl elements in the Kashiwara-Tanisaki character theorem [KT00] which correspond to the weights of the singular vectors.
We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.
We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q,t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q,t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defi…
We construct a new family of flat connections generalising the KZ connection, the Casimir connection and the dynamical connection. These new connections are attached to simply-laced graphs, and are obtained via quantisation of time-dependent Hamiltonian systems controlling the isomonodromic deformations of meromorphic connections on the sphere.
Dorey's Rule and the q-Characters of Simply-Laced Quantum Affine Algebras
设Uq（ghat）是单缀层单李代数g的量子仿射代数.我们研究Dorey规则，这是一个几何陈述的Coxeter轨道的g-重量，和结构的q-字符的基本表示V_{i,a}的Uq（ghat）之间的关系。特别是，我们证明，没有求助于ADE分类，该规则提供了一个充要条件单项式1出现在一个三重张量积V_{i,a} x V_{j,b} x V_{k,c}的q-字符。
We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group $\mathbf{U}_{\mathfrak{q}\tilde{\mathfrak{q}}}(\mathfrak{g}_\mathbb{R})$ of the modular double …
Generating Functions with $τ$-Invariance and Vertex Representations of Quantum Affine Algebras $U_{r,s}(\widehat{\mathfrak{g}})$ (I): Simply-laced Cases
We prove that the Grothendieck rings of category $\mathcal{C}^{(t)}_Q$ over quantum affine algebras $U_q'(\g^{(t)})$ $(t=1,2)$ associated to each Dynkin quiver $Q$ of finite type $A_{2n-1}$ (resp. $D_{n+1}$) is isomorphic to one of category $\mathcal{C}_{\mQ}$ over the Langlands dual $U_q'({^L}\g^{(2)})$ of $U_q'(\g^{(2)})$ associated to any twisted adapted class $[\mQ]$ of $A_{2n-1}$ (resp. $D_{n+1}$). This results provide partial answers of conjectures of Frenkel-Hernandez on Langlands duality for finite-dimensional representation of quantum affine algebras.
In this paper, we seek to prove the equality of the $q$-graded fermionic sums conjectured by Hatayama et al. in its full generality, by extending the results of Di Francesco and Kedem to the non-simply laced case. To this end, we will derive explicit expressions for the quantum $Q$-system relations, which are quantum cluster mutations that correspond to the classical $Q$-system relations, and write the identity of the $q$-graded fermionic sums as a constant term identity. As an application, we will show that these quantum $Q$-system relations are consistent with the short exact sequence of the…
We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $Ψ$-system among subdominant solutions. We then use the $Ψ$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explici…
We construct the positive principal series representations for $U_q(g_R)$ where $g$ is of type $B_n$, $C_n$, $F_4$ or $G_2$, parametrized by $R^r$ where $r$ is the rank of $g$. We show that under the representations, the generators of the Langlands dual group $U_{\tilde{q}}({}^Lg_R)$ are related to the generators of $U_q(g_R)$ by the transcendental relations. We define the modified quantum group of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the $q$-tori polynomials.
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $\mathscr{C}_{\mathfrak{g}}$ be the category of finite-dimensional integrable $U_q'(\mathfrak{g})$-modules. We write $\mathscr{C}_{\mathfrak{g}}^0$ for the monoidal subcategory of $\mathscr{C}_{\mathfrak{g}}$ introduced by Hernandez-Leclerc. In this paper, we associate a simply-laced finite type root system to each quantum affine algebra $U_q'(\mathfrak{g})$ in a natural way, and show that the block decompositions of $\mathscr{C}_{\mathfrak{g}}$ and $\mathscr{C}_{\mathfrak{g}}^0$ are parameterized by the lat…
Using vertex operators, we build representations of the Yangian of a simply laced Kac-Moody algebra and of its double. As a corollary, we prove the PBW property for simply laced affine Yangians.
We prove an explicit inverse Chevalley formula in the equivariant $K$-theory of semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley formula, we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb{Z}[q^{\pm 1}]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply-laced type and equivariant scalars $e^λ$, where $λ$ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines …
The spectrum of integrable models is often encoded in terms of commuting functions of a spectral parameter that satisfy functional relations. We propose to describe this commutative algebra in a covariant way by means of the extended Q-system that comprise Q-vectors in each of the fundamental representations of the (Langlands dual of) the underlying symmetry algebra. These Q-vectors turn out to parameterise a collection of complete flags which are fused with one another in a particular way. We show that the fused flag is a finite-difference oper in a particular gauge, explicit identification d…
Let g be a semi-simple Lie algebra. In this paper we study the spaces of based quasi-maps from the projective line P^1 to the flag variety of g (it is well-known that their singularities are supposed to model the singularities of the so called semi-infinite Schubert varieties which are hard to define directly). In the first part of the paper we show that the above spaces are normal and in the case when g is simply laced they are also Gorenstein and have rational singularities. In the second part of the paper we compute the character of the ring of functions on the above spaces; in view of the …
The main result is that the category of ordinary modules of an affine vertex operator algebra of a simply laced Lie algebra at admissible level is rigid and thus a braided fusion category. If the level satisfies a certain coprime property then it is even a modular tensor category. In all cases open Hopf links coincide with the corresponding normalized S-matrix entries of torus one-point functions. This is interpreted as a Verlinde formula beyond rational vertex operator algebras. A preparatory Theorem is a convenient formula for the fusion rules of rational principal W-algebras of any type.
In this paper we construct combinatorial bases of parafermionic spaces associated with the standard modules of the rectangular highest weights for the untwisted affine Lie algebras. Our construction is a modification of G. Georgiev's construction for the affine Lie algebra $\widehat{\mathfrak sl}(n+1,\mathbb C)$---the constructed parafermionic bases are projections of the quasi-particle bases of the principal subspaces, obtained previously in a series of papers by the first two authors. As a consequence we prove the character formula of A. Kuniba, T. Nakanishi and J. Suzuki for all non-simply-…
