# Algebra of operators for Heisenberg XXZ spin chain model

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Recently I've been reading Reshetikhin's lecture note https://arxiv.org/pdf/1010.5031.pdf on integrability of the 6-vertex model. The author defines a complex algebra $C_q(\widehat{\text{SL}}_2)$ as the following.

Consider the $R$-matrix

$$R=\begin{bmatrix} 1 & 0 & 0& 0\\ 0 & f(z) & z^{-1}g(z) & 0\\ 0 & zg(z) & f(z) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

where

$$f(z)=\frac{z-z^{-1}}{zq-z^{-1}q^{-1}},\quad g(z)=\frac{q-q^{-1}}{zq-z^{-1}q^{-1}}.$$

Consider the matrix $\mathcal{T}(z)$ which is the generating function for the elements $T^{(k)}_{ij}$

$$\mathcal{T}(z)=\sum^{\infty}_{k=1}T^{(k)}z^{2k}+\begin{bmatrix} T^{(0)}_{11} & T^{(0)}_{12}\\ 0 & T^{(0)}_{22} \end{bmatrix},$$

where $T^{(k)}_{ij}$ is a matrix element of $T^{(k)}$ for $k\ge 1$. Then the defining relations of $C_q(\widehat{\text{SL}}_2)$ can be written as the following matrix identities with entries in $C_q(\widehat{\text{SL}}_2)$:

$$R(z)\mathcal{T}(zw)\otimes \mathcal{T}(w)=(1\otimes \mathcal{T}(w))(\mathcal{T}(zw)\otimes 1)R(z)$$

and

$$\mathcal{T}(qz)_{11}\mathcal{T}(z)_{22}- \mathcal{T}(qz)_{12}\mathcal{T}(z)_{21}=1.$$

I have a feeling that this definition of $C_q(\widehat{\text{SL}_2})$ might have something to do with the affine quantum group $U_q(\widehat{\text{SL}_2})$, but I am not sure at this moment. Are they the same, or if not how are they related to each other?

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The complex algebra Cq(SL2) defined by Reshetikhin is related to the affine quantum group Uq(SL2) as follows:

Cq(SL2) is a specialization of the quantized enveloping algebra Uq(sl2), which is the quantization of the Lie algebra sl2. This means that Cq(SL2) can be obtained from Uq(sl2) by setting certain parameters to specific values.

Uq(SL2) is the quantum group associated with the Lie algebra sl2 and can be thought of as a deformation of the Lie group SL2. It is a Hopf algebra and has several important structures and properties, such as the R-matrix, the coproduct, and the antipode.

In the case of Cq(SL2), the R-matrix is defined in terms of a function f(z) and g(z) and the matrix elements T(k)ij. The defining relations of Cq(SL2) are then given by the matrix identities involving T(z) and R(z).

In summary, Cq(SL2) is a specialization of Uq(sl2) and is related to the affine quantum group Uq(SL2) as a subalgebra or a quotient algebra.

answered Feb 6, 2023 by anonymous

Hi, thanks for the answer. Do you have a reference for that? I would like to check the precise statement.

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