Given a semisimple Lie algebra, we can construct subalgebras corresponding to removing nodes from the extended Dynkin diagram. For example $\mathfrak{su}_3 \subseteq \mathfrak{g}_2$. I'm interested in classifying how these subalgebras deform when we pass to the quantum group $U_q (\mathfrak{g})$. It doesn't need to respect the Hopf algebra structure in any way, and it only needs to behave in the expected way on the Cartan subalgebra $\frak h$. Are there results from the literature discussing this?
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