Quantizing solutions to the reflection algebra

Question

I am trying to find the quantum analogues to classical solutions of Sklyanin's reflection algebra (RE). I have a solution to the classical Poisson bracket for known r-matrix $r(\mu)$ \begin{equation}\tag{RE} \{\overset{1}K(\mu),\overset{2}K(\lambda)\}=[r(\mu-\lambda),\overset{1}K(\mu)\overset{2}K(\lambda)]+\overset{1}K(\mu)r(\mu+\lambda)\overset{2}K(\lambda)-\overset{2}K(\lambda)r(\mu+\lambda)\overset{1}K(\mu), \end{equation}

and wish to find a solution corresponding to the same physical system quantized. According to Sklyanin's paper (http://iopscience.iop.org/article/10.1088/0305-4470/21/10/015/meta), I believe I should be looking for solutions to

\begin{align}\tag{QRE} R(\mu-\lambda)\overset{1}K(\mu)R(\mu+\lambda)\overset{2}K(\lambda)=\overset{2}K(\lambda)R(\mu+\lambda)\overset{1}K(\mu)R(\mu-\lambda). \end{align}

I have directly tried with the same solutions (and the transformation $[\hspace{.2cm},\hspace{.2cm}]=-i\hbar \{\hspace{.2cm},\hspace{.2cm}\}$), and it agrees up to order $\hbar$ as expected by the semiclassical approximation (limit) $\hbar\rightarrow 0$ and $R(\mu)=1+i\hbar r(\mu)+O(\hbar^2)$, but not to higher order (which my general algebraic approach says is correct). My only other experience with a similar procedure is quantizing the periodic Toda chain with the two equations being $\{\overset{1}T(\mu),\overset{2}T(\lambda)\}=[r(\mu-\lambda),\overset{1}T(\mu),\overset{2}T(\lambda)]$ and $R(\mu-\lambda)\overset{1}T(\mu)\overset{2}T(\lambda)=\overset{2}T(\lambda)\overset{1}T(\mu)R(\mu-\lambda)$ and the solutions $T(\mu)$ worked in both cases with $R(\mu)=1+i\hbar r(\mu)$ so I'm at a bit of a loss of how to proceed.

My intuition is telling me that maybe I need to shift my classical solutions spectral parameter, but I'm unsure if that works. I've also found material on the other direction (https://arxiv.org/abs/hep-th/9209054) which takes a quantum group and shows the semiclassical approximation will give the classical equation (and hence the solutions), but I need to go the other way.

Sorry for rambling I'm pretty stuck and couldn't find any material anywhere. Even a simple example of quantizing the RE algebra for a simple model in which the solutions change would be really helpful!

Answer 1

Here are some comments to help you on the way.

Your (QRE) is almost correct: on both sides of the equality the second $R$-matrix should in general be $R_{21} = P \, R \, P$ with $P$ the permutation. (So, by definition, you don't see this for the case of a symmetric $R$-matrix.)

This change can be understood in terms of the usual graphical notation. For definiteness let's think of time as increasing upwards. The $K$-matrix can be depicted as a sort of "K", with the | depicting the reflecting end, which we think of as a wall or mirror on the left, and the rule for $K(\mu)$ is that the 'incoming' (from the right) spectral parameter is $-\mu$, which is reflected to the 'outgoing' (back to the right) parameter $+\mu$. The sign makes sense if we would interpret the spectral parameters as slopes, cf the usual interpretation of the Yang--Baxter equation (YBE) as factorised scattering for 2d integrable QFT. The (quantum) reflection equation then says that the two $K$s --- one above the other on the same wall --- can be slid through each other, rather like the YBE graphically is about sliding one line through the crossing of the two other lines. The arguments of the two $R$-matrices in the reflection equation can be understood as well: as for the YBE, the rule is that the $R$-matrix with incoming lines with parameters $\lambda$ on the left and $\mu$ on the right has arguments $R(\lambda-\mu)$, and the parameters follow the lines.

The resulting equation is also known as the boundary Yang--Baxter equation, and its solutions have been studied to quite some extent. This should at least be enough for you to find various examples, e.g. for the six-vertex model (XXZ spin chain) with either 'diagonal' or generic reflection.