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。
英文摘要：
Let Uq(ghat) be the quantum affine algebra associated to a simply-laced simple Lie algebra g. We examine the relationship between Dorey's rule, which is a geometrical statement about Coxeter orbits of g-weights, and the structure of q-characters of fundamental representations V_{i,a} of Uq(ghat). In particular, we prove, without recourse to the ADE classification, that the rule provides a necessary and sufficient condition for the monomial 1 to appear in the q-character of a three-fold tensor product V_{i,a} x V_{j,b} x V_{k,c}.
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 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.
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…
Generating Functions with $τ$-Invariance and Vertex Representations of Quantum Affine Algebras $U_{r,s}(\widehat{\mathfrak{g}})$ (I): Simply-laced Cases
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 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 …
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.
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 …
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 …
We introduce quiver gauge theory associated with the non-simply-laced type fractional quiver, and define fractional quiver W-algebras by using construction of arXiv:1512.08533 and arXiv:1608.04651 with representation of fractional quivers.
In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.
In this note, we extend the results of arxiv:1111.2266 and arxiv:1203.1583 to the non simply laced case. To this end we introduce and study the twisted zastava spaces.
