First of all, my question is motivated by the following phenomenon. Suppose that $Y$ is a quantisation of a Lie bialgebra (say $y$), and suppose that $Y' \subset Y$ be a coideal subalgebra. In the theory of quantum symmetric pairs, $Y'$ is usually given by an RTT-style presentation (and I believe that such an RTT-style presentation is needed for establishing $Y'$ as a quantisation of some Lie coideal $y' \subset y$). In general, $Y'$ may not admit a triangular decomposition.
What is interesting/strange is that $Y'$ - despite not being triangular - may sometimes still admit a presentation of the Drinfeld-Chevalley-Serre type, i.e. in terms of raising/lowering operators and Cartan-like "diagonal" operators. For example, we do know by a recent work by Lu-Zhang-Wang on small-rank split-type twisted Yangians (and its sequels) that for those algebras, such a Drinfeld-Chevalley-Serre presentation exists, but as far as I know the problem is still open in general. Despite this, we do have the notion of highest-weight $Y'$-modules. For instance, take $Y$ to be the Yangian of some $gl_n$ and $Y'$ to be a twisted Yangian therein, then it is even possible to classify the finite-dimensional simple $Y'$-modules, and this result as well as the techniques used to prove it bear many similarities to Drinfeld's classification of finite-dimensional simple $Y(gl_n)$-modules. In these examples, however, we rely heavily on the fact that the relations defining $Y'$ are "deformations" of the commutation relations between matrices in the underlying classical Lie algebra: for instance, we can use upper/lower-triangular matrices as raising/lowering operators.
This prompts the following question from me: has there ever been any attempt at establishing a highest-weight theory using only the RTT relations ?
Thank you for any comment and/or answer!
