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Let $U_q(\frak{sl}_2)$ denote the quantum universal enveloping algebra of $\frak{sl}_2$, and consider the adjoint action $$ \mathrm{ad}_X: U_q({\frak sl}_2) \to U_q({\frak sl}_2), ~~ Y \mapsto S(X_{(1)})YX_{(2)}, $$ where we have used sumless Sweedler notation. This gives $U_q({$\frak sl}_2)$ the structure of a $U_q({\frak sl}_2)$-module. What is the structure of this module? Does it decompose into a direct sum of finite-dim irreps, or are there infinite-dim reps in there? If decomposes into a direct sum of finite-dim irreps, then does every irrep appear, and are there multiplicities. How does it compare to the classical situation?

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  • $\begingroup$ The quantized enveloping algebra is finitely generated but infinite-dimensional, so that answers at least some of your questions negatively. Have you looked in Jantzen's quantum groups book? If not, I think you'd find it very helpful, re how modules for U_q look. $\endgroup$ Jul 9, 2021 at 15:41
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    $\begingroup$ Sorry but I don't see which questions are answered negatively by the fact that $U_q({\frak sl}_2)$ is an infinite-dim space. One can easily produce an infinite-dim $U$-module which decomposes into a direct sum of infinitely many fin-dim submodles. $\endgroup$ Jul 9, 2021 at 16:33
  • $\begingroup$ Personally I wouldn't say "direct sum" in this context to include a case with infinitely many factors, but perhaps I've spent too long with fin dim algebras people. Re the question itself, have you looked at the subspace spanned by <H,H^{-1}>? $\endgroup$ Jul 10, 2021 at 9:48

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Unlike what happens in the classical case, it is not locally finite dimensional, basically because it has invertible elements. However its ad-locally finite part $U'$ is very large and it decomposes as a direct sum $$U'=\bigoplus_{V} V^* \otimes V$$ where the sum is over the irreducible finite dimensional modules (note this is a purely 'quantum' phenomena).

A standard reference is Joseph-Letzter "Local finiteness of the adjoint action for quantized enveloping algebras".

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  • $\begingroup$ In what sense do you mean "dense"? $\endgroup$ Jul 13, 2021 at 12:19
  • $\begingroup$ Also - I guess the sum is multiplicity free? $\endgroup$ Jul 13, 2021 at 12:21
  • $\begingroup$ Sorry, dense wasn't the right word here, I'll edit. A precise sense in which it's large is explained in the linked paper. And yes, the sum is multiplicity free. $\endgroup$
    – Adrien
    Jul 13, 2021 at 12:38

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