A link between two elliptic quantum groups

The Center for the Elliptic Quantum Group $E_{τ,η}(sl_n)$

We give the center of the elliptic quantum group in general case. Based on the Dynamic Yang-Baxter Relation and the fusion method, we prove that the center commute with all generators of the elliptic quantum group. Then for a kind of assumed form of these generators, we find that the coefficients of these generators form a new type closed algebra. We also give the center for the algebra.

Solutions of the qKZB Equation in Tensor Products of finite dimensional modules over the elliptic quantum group $E_{τ,η}sl_2$

We consider the quantized Knizhnik-Zamolodchikov-Bernard difference equation (qKZB) with step $p$ and values in a tensor product of finite dimensional evaluation modules over the elliptic quantum group $E_{τ,η}(sl_2)$, the equation defined in terms of elliptic dynamical R-matrices. We solve the equation in terms of multidimensional q-hypergeometric integrals and describe its monodromy properties. We identify the space of solutions of the qKZ equation with the space of functions with values in the tensor product of the corresponding modules over the elliptic quantum group $E_{p,η}(sl_2)$.

Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations

The elliptic quantum Knizhnik-Zamolodchikov-Bernard (qKZB) difference equations associated to the elliptic quantum group $E_{τ,η}(sl_2)$ is a system of difference equations with values in a tensor product of representations of the quantum group and defined in terms of the elliptic R-matrices associated with pairs of representations of the quantum group. In this paper we solve the qKZB equations in terms of elliptic hypergeometric functions and decribe the monodromy properties of solutions. It turns out that the monodromy transformations of solutions are described in terms of elliptic R-matrice…

Algebraic Bethe ansatz for the elliptic quantum group $E_{τ,η}(sl_2)$

To each representation of the elliptic quantum group $E_{τ,η}(sl_2)$ is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this construction give eigenvectors for IRF models, for the eight-vertex model and for the two-body Ruijsenaars operator. The latter is a $q$-deformation of Hermite's solution of the Lamé equation.

Elliptic quantum groups and Ruijsenaars models

We construct symmetric and exterior powers of the vector representation of the elliptic quantum groups $E_{τ,η}(gl_N)$. The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested Bethe ansatz method. In special cases we recover the Ruijsenaars systems of commuting difference operators.

椭圆量子群

Elliptic Quantum Groups

本研究阐述Drinfeld实现下关联仿射李代数与环面代数的椭圆量子群，统一不同级别表示并定义顶点算子，建立其0级表示与等变椭圆上同调的对应，明确其作为W代数q形变的特征。

Conjugacy classes in loop groups and G-bundles on elliptic curves

We added an additional result (theorem 1.6) that strengthenns our main theorem in the G=GL-case by establishing an equivalence of tensor categories.

Elliptic Stable Envelopes and Finite-dimensional Representations of Elliptic Quantum Group

We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we…

Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra U_{q,p}(\hat{sl}_2). The algebra U_{q,p}(\hat{sl}_2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra U_q(\hat{sl}_2) and a Heisenberg algebra. This yields a systematic construction of both finite and infinite-dimensional dynamical representations and their parallel structures to U_q(\hat{sl}_2). In particular we give a classification theorem of the finite-dimensional irreducible pseudo-highest weight represe…

Creation operators and algebraic Bethe ansatz for the elliptic quantum group $E_{τ,η}(so_3)$

We define the elliptic quantum group $E_{τ,η}(so_3)$ and the transfer matrix corresponding to its simplest highest weight representation. We use Bethe anstaz method to construct the creation operators as polynomials of the Lax matrix elements expressed through a recurrence relation. We give common eigenvectors and eigenvalues of the family of commuting transfer matrices.

Surface defects and elliptic quantum groups

A brane construction of an integrable lattice model is proposed. The model is composed of Belavin's R-matrix, Felder's dynamical R-matrix, the Bazhanov-Sergeev-Derkachov-Spiridonov R-operator and some intertwining operators. This construction implies that a family of surface defects act on supersymmetric indices of four-dimensional $\mathcal{N} = 1$ supersymmetric field theories as transfer matrices related to elliptic quantum groups.

Construction of the Bethe State for the $E_{τ,η}(so_3)$ Elliptic Quantum Group

Elliptic quantum groups can be associated to solutions of the star-triangle relation of statistical mechanics. In this paper, we consider the particular case of the $E_{τ,η}(so_3)$ elliptic quantum group. In the context of algebraic Bethe ansatz, we construct the corresponding Bethe creation operator for the transfer matrix defined in an arbitrary representation of $E_{τ,η}(so_3)$.