Solution of the Yang-Baxter equation associated to the $U_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra

Question

I have a solution (a $R$ matrix) of the Yang-Baxter equation, \begin{equation} R_{12}(x_{1})R_{13}(x_{1}x_{2})R_{23}(x_{2})=R_{23}(x_{2})R_{13}(x_{1}x_{2})R_{12}(x_{1}) \end{equation}

that probably is associated to the $ U_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra, but I'm not sure. I would like to certify myself about this. Anybody know how can I proceed?

I know that $ U_q[osp(2n+2|2m)^{(2)}]$ is the orthosymplectic twisted Lie superalgebra, $r=2n+2$ being the number of bosons and $2m$ the number of fermions, however it seems to be very difficult to find any reference of it. The only reference I found, after an extensive search, was the paper of Frappat et Al. named "Structure of basic Lie superalgebras and of their affine extensions", (Commun. Math. Phys. 121,457-500 ,1989), were some information about twisted Lie algebras was given and I found as well the Dynkin diagrams for the $ U_q[osp(2n+2|2m)^{(2)}]$.

Moreover, for $m=0$ it seems that the Lie superalgebra $ U_q[osp(2n+2|2m)^{(2)}]$ reduces to the $U_q[D_{n+1}^{(2)}]$ classical Lie algebra. In fact, I checked that this $R$ matrix reduces to the Jimbo's $R$ matrix for $m=0$ which is associated to the $U_q[D_{n+1}^{(2)}]$ Lie algebra. I'm also interested in the special case $n=0$, that is, the $ U_q[osp(2|2m)^{(2)}]$ Lie algebra.

Can anyboby give some expanation of these issues, or indicate some reference related to? I will be very tankful for any help. Thank you in advance!

Answer 1

Drinfel'd's quantum double is a construction that produces, given a Hopf algebra, an $R$-matrix that turns it into a quasi-triangular Hopf algebra. You could try working that out to get an $R$-matrix for the $U_q(\mathfrak{osp}(...|...))$ that you propose, and see if it matches with what you hope to find.

For this it might be instructive to see how the construction works for Lie superalgebras by looking at the example of $\mathfrak{sl}(2|2)$ (and $\mathfrak{psu}(2,2|4)$), which has received quite a lot of attention in the past decade in the context of the AdS/CFT correspondence: see Beisert, arXiv:1602.04988.