Two FRT bialgebroids and their relations

FRT Construction for Dynamical Yang-Baxter Maps

Notions of an (H, X)-bialgebroid and of its dynamical representation are proposed. The dynamical representations of each (H, X)-bialgebroid form a tensor category. Every dynamical Yang-Baxter map R(lambda) satisfying suitable conditions, a generalization of the set-theoretical solution to the quantum Yang-Baxter equation, gives birth to an (H, X)-bialgebroid A_R. The categories of L-operators for R(lambda) and of dynamical representations of A_R are isomorphic as tensor categories.

Tannaka theory and the FRT construction over non-commutative algebras

Let $A$ be an algebra over a commutative ring $k$. We introduce the notion of a coquasitriangular left bialgebroid over $A$ and show that the category of left comodules over such a bialgebroid has a braiding. We also investigate a Tannaka type construction of bimonads and bialgebroids. As an application, the Faddeev-Reshetikhin-Takhtajan (FRT) construction over the algebra $A$ is established. Our construction associates a coquasitriangular bialgebroid to a braided object $(M, c)$ in the category of $A$-bimodules such that $M$ is finitely generated and projective as a left $A$-module. A Hopf al…

两类FRT型量子顶点代数

On two families of quantum vertex algebras of FRT-type

本文提出两类与A型三角、八顶点模型椭圆R-矩阵相关的FRT型量子顶点代数，证明其φ-坐标表示由FRT算子支配并展示应用，还研究了椭圆情形下量子行列式的性质。

Quiver-theoretical approach to dynamical Yang-Baxter maps

A dynamical Yang-Baxter map, introduced by Shibukawa, is a solution of the set-theoretical analogue of the dynamical Yang-Baxter equation. In this paper, we initiate a quiver-theoretical approach for the study of dynamical Yang-Baxter maps. Our key observation is that the category of dynamical sets over a set $Λ$, introduced by Shibukawa to establish a categorical framework to deal with dynamical Yang-Baxter maps, can be embedded into the category of quivers with vertices $Λ$. By using this embedding, we shed light on Shibukawa's classification result of a certain class of dynamical Yang-Baxte…

Quantum groups obtained from solutions to the parametrized Yang-Baxter equation

In this article we use a parametrized version of the FRT construction to construct two new coquasitriangular Hopf algebras. The first one, $\widehat{SL_q(2)}$, is a quantization of the coordinate ring on affine $SL(2)$. We show that there is a duality relation between this object and the more well-known $U_q(\widehat{\mathfrak{sl}_2})$. We then build certain irreducible comodules of this Hopf algebra and prove an irreducibility criterion for their tensor product in the spirit of Chari and Pressley. The second object is built from a solution of the parametrized Yang-Baxter equation with paramet…