We work over the field of complex numbers. We have known that Lie algebra of type $A_2 $is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there exist $p,q$ such that $U_q(\mathfrak{sl}(3))$ is a subalgebra of $U_p(\mathfrak{g})$ as associative algebra, where $\mathfrak{g}= G_2$?
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$\begingroup$ Since this algebra also have a coalgebra structure, and I just need to consider it as an algebra. $\endgroup$– user11090426Commented Nov 20, 2019 at 0:55
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$\begingroup$ Since they are given by generators and relations, why not just to check the relations for $U_q(sl(3))$ on "long root" generators of $U_p({\mathfrak g})$? Or do you need a more "functorial" answer for further applications? $\endgroup$– Victor PetrovCommented Nov 20, 2019 at 17:23
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