Two-parameter Quantum Affine Algebra $U_{r,s}(\widehat{\frak {sl}_n})$, Drinfel'd Realization and Quantum Affine Lyndon Basis

On a class of representations of the Yangian and moduli space of monopoles

A new class of infinite dimensional representations of the Yangians $Y(\frak{g})$ and $Y(\frak{b})$ corresponding to a complex semisimple algebra $\frak{g}$ and its Borel subalgebra $\frak{b}\subset\frak{g}$ is constructed. It is based on the generalization of the Drinfeld realization of $Y(\frak{g})$, $\frak{g}=\frak{gl}(N)$ in terms of quantum minors to the case of an arbitrary semisimple Lie algebra $\frak{g}$. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces…

Nichols algebras over classical Weyl groups, Fomin-Kirillov algebras and Lyndon basis

We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over the classical Weyl groups are infinite dimensional. We establish the relationship between Fomin-Kirillov algebra $\mathcal E_n$ and Nichols algebra $\mathfrak{B} ({\mathcal O}_{(1, 2)} , ε\otimes {\rm sgn})$ of transposition over symmetry group by means of quiver Hopf algebras. We generalize {\rm FK } algebra. The characteristic of finiteness of Nichols alge…

$\frak{g}$-quasi-Frobenius Lie algebras

A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for $\frak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\frak{q},β)$ together with a left $\frak{g}$-module structure which acts on $\frak{q}$ via derivations and for which $β$ is $\frak{g}$-invariant. Geometrically, $\frak{g}$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie grou…

Quantization of Poisson groups -- II

Let $ G^τ$ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfeld structure of Poisson group; let $ H^τ$ be its dual Poisson group. By means of Drinfeld's double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,ϕ}^M ({\frak h}) $ --- dual of $ U_{q,ϕ}^{M'} ({\frak g}) $ --- which yield infinitesimal quantization of $ H^τ$ and $ G^τ$; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $…

Zeros and poles of quantum current operators and the condition of quantum integrability

For the current realization of the affine quantum groups, a simple comultiplication for the quantum current operators was given by Drinfeld. With this comultiplication, we study the zeros and poles of the quantum current operators and present a condition of integrability on the quantum current of $U_q\left(\hat{\frak sl}(2)\right)$, which is a deformation of the corresponding condition for $\hat{\frak sl}(2)$. We also present the results about the zeros and poles of the quantum current operators of $U_q\left(\hat{\frak sl}(n)\right)$.

Yang-Baxter Equations for General Metaplectic Ice

In this paper, we extend results connecting quantum groups to spherical Whittaker functions on metaplectic covers of $GL_r(F)$, for $F$ a nonarchimedean local field. Brubaker, Buciumas, and Bump showed that for a certain metaplectic $n$-fold cover of $GL_r(F)$ a set of Yang-Baxter equations model the action of standard intertwiners on principal series Whittaker functions. These equations arise from a Drinfeld twist of the quantum affine Lie superalgebra $U_{\sqrt{v}}(\widehat{\frak{gl}}(n)),$ where $v = q^{-1}$ for $q$ the cardinality of the residue field. We extend their results to all metapl…

Difference equations of quantum current operators and quantum parafermion construction

For the current realization of the affine quantum groups, a simple comultiplication for the quantum current operators was given by Drinfeld. With this comultiplication, we prove that, for the integrable modules of $U_q(\hat {\frak sl}(2))$ of level $k+1$, $x^\pm(z)x^\pm(zq^{\pm 2}) \cdot\cdot\cdot x^\pm(zq^{\pm 2k})$ are vertex operators satisfying certain q-difference equations, and we derive the quantum parafermions of $U_q(\hat {\frak sl}(2))$.

量子对称空间

Quantum symmetric spaces

设$G$是一个半单李群，${\frak g}$是它的李代数.对于$G$上的对称空间$M$，我们构造了$M$上光滑函数空间$A$中的一个新的（变形）乘法.这个乘法在Drinfeld-Jimbo量子群$U_h{\frak g}$的作用下是不变的，并且关于对合算子$\tilde{S}: A\otimes A \to A\otimes A$是可交换的。这种倍增是独一无二的。设$M$是一个具有标准Poisson结构的Kähler对称空间.然后我们在$A$中构造了一个依赖于两个参数的$U_h{\frak g}$ -不变乘法，它是该结构的量化。

Quantization of Poisson Groups

Let $ G^τ$ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group, let $ H^τ$ be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,φ}^M(\frak{h}) $ --- dual of $ U_{q,φ}^{M'}(\frak{g}) $ --- which yield infinitesimal quantization of $ H^τ$ and $ G^τ\, $, we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $ H^…

两参数量子仿射代数$U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n+1}})$的Ding-Frenkel同构定理

The Ding-Frenkel Isomorphism Theorem for two-parameter quantum affine algebra $U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n+1}})$

本文基于有限维权模理论，研究了两参数量子仿射代数的量子Lyndon基构造及其在三角$L$-矩阵中的分布规则，并给出了Drinfeld-Jimbo表示与FRT表示之间同构性的代数证明。

Lusztig 量子根向量与 A 系列全量子旗流形的 Dolbeault 复形

Lusztig's Quantum Root Vectors and a Dolbeault Complex for the A-Series Full Quantum Flag Manifolds

本文证明 Lusztig 正根向量在好约化分解下构成 A 系列全量子旗流形的量子切空间，其微分演算为经典反全纯 Dolbeault 复形的 q-形变，并由此建立量子 Borel-Weil 定理，实现所有有限维 type-1 不可约表示。

受限杨代数的洗牌代数实现

Shuffle algebra realizations for modular Yangians

本文研究了在特征p>2的域上，sl_n的杨代数正子代数Y_n^(>)(k)的洗牌代数实现，确定了其核和像，并揭示了与特征p相关的轮条件。