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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!

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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.

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  • $\begingroup$ Thanks! I've looked up at classifying the solutions to this for XXX/six-vertex model which is useful. I actually am doing it for the Toda lattice, and think I need my boundary matrices to be dynamical (hence entries to be non-commuting) has any work been done in this direction to your knowledge? $\endgroup$ – Nick Macleod Jan 31 at 6:48
  • $\begingroup$ @NickMacleod Yup, the dynamical case (in the sense of the dynamical YBE, with the shifts of the dynamical parameters depending on the weight) has also been studied for the elliptic SOS, aka 8VSOS, or IRF model (of which the six-vertex model is a special case). Look for "dynamical K-matrix". From the top of my head: see papers of Behrend Pearce O'Brien (in the IRF language) and of Stokman (more mathematical) $\endgroup$ – Jules Lamers Feb 1 at 7:13

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