The $R$-matrix of the quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ in the Fock module

$(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody $U_q(\widehat{\mathfrak{g}})_k$ to generic quantum toroidal algebras. A nearly exhaustive presentation is given for the two series $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$ and $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_n)$, when screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations. Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (…

Finite type modules and Bethe Ansatz for quantum toroidal gl(1)

We study highest weight representations of the Borel subalgebra of the quantum toroidal gl(1) algebra with finite-dimensional weight spaces. In particular, we develop the q-character theory for such modules. We introduce and study the subcategory of `finite type' modules. By definition, a module over the Borel subalgebra is finite type if the Cartan like current ψ^+(z) has a finite number of eigenvalues, even though the module itself can be infinite dimensional. We use our results to diagonalize the transfer matrix T_{V,W}(u;p) analogous to those of the six vertex model. In our setting T_{V,W}…

广义麦克唐纳函数与量子环面$\boldsymbol{\rm gl(1)}$代数

Generalized Macdonald functions and quantum toroidal gl(1) algebra

本研究将普通Macdonald对称函数的$e_1$-Pieri规则、与Whittaker向量相关恒等式等公式推广到量子环面$\rm \frak{gl}(1)$代数对应的广义情形，提出五项关系与Fourier/Hopf配对的推广，并证明了广义Macdonald核的因子化猜想。

量子扭环$\mathfrak{gl}_1$共轭对的扩展

Extensions of a commuting pair of quantum toroidal $\mathfrak{gl}_1$

本文提出一族代数$\mathcal{A}_{M,N}$作为量子扭环$\mathfrak{gl}_1$子代数的扩展，探讨其coproduct同态结构，并在Fock模块上构造了相关模的例子。

Quantum $W_{1+\infty}$ subalgebras of BCD type and symmetric polynomials

The infinite affine Lie algebras of type ABCD, also called $\widehat{\mathfrak{gl}}(\infty)$, $\widehat{\mathfrak{o}}(\infty)$, $\widehat{\mathfrak{sp}}(\infty)$, are equivalent to subalgebras of the quantum $W_{1+\infty}$ algebras. They have well-known representations on the Fock space of either a Dirac fermion ($\hat A_\infty$), a Majorana fermion ($\hat B_\infty$ and $\hat D_\infty$) or a symplectic boson ($\hat C_\infty$). Explicit formulas for the action of the quantum $W_{1+\infty}$ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of …

喷流空间中的扩张双同态代数和轨迹

Extended diffeomorphism algebras and trajectories in jet space

设DRO代数DRO（N）是$diff(N)\oplus diff(1)$通过其四个不等价的Virasoro-like上圈而扩张的.这里$diff(N)$是$N$维时空中的同胚代数，而$diff(1)$描述了张量值$p$ -喷流空间中轨迹的重新参数化。DRO（N）具有用于每个$p$和$gl(N)$的每个表示的Fock模块。给出了规范代数（高维卡茨-穆迪代数）的类似表示.通过规范固定过程可以消除重新参数化对称性，从而得到先前发现的模块。在这个过程中，两个DRO（N）余圈转化为$diff(N)$的各向异性余圈。从而从几何上解释了环面李代数及其导子代数的Fock模。

二维Toda可积层级的一种$(q,t)$形变

A $(q,t)$-deformation of the 2d Toda integrable hierarchy

本研究通过将对称代数提升为量子环面$\u27e8gl(1)$代数，构造出二维Toda可积层级的$(q,t)$形变，推导其双线性结构与多tau函数方程，定义普适细化tau函数并得到类$RLL$的二次方程，丰富了可积系统的形变理论。