Question:

Hopf Subalgebras of Quantized Algebras

Lincoln: 3 days ago

As is well known, quantized enveloping algebras $U_q(\frak{g})$ admit far fewer sub-Hopf algebras than classical enveloping algebras $U(\frak{g})$. As one can check directly, for appropriate subsets of the (simple root) standard generators $E_i,F_i,K_i$, a Hopf subalgebras are seen to be generated. However, I do not know of sub-Hopf algebras which are not generated by the standard generators. Are there examples of objects? Moreover, since the notion of a Hopf sub-algebra is so restrictive for $U_q(\frak{g})$, is it naive to expect some kind of classification?

Answer:
Weston: 2 days ago

Since we know from Etingof-Kazhdan that quantization is functorial we can safely say that the classification of sub-Lie bialgebra, which was obtained in the standard case, implies classification of sub-Hopf algebras.

The paper in which classification of Lie bialgebras is obtained is J. Stokman in "the quantum orbit method for generalized flag manifold" arxiv: math/0206245, Proposition 2.1 and basically the result says that yes, you just get subLie bialgebras standardly generated.

Coideal subalgebras, as suggested by Grabowski, are a much richer family; stil some classification results were obtained.