# Isomorphism Between $U_{h}(\frak{sl}_3)$ and $U({\frak sl}_3)[[h]]$

+ 4 like - 0 dislike
317 views

I take the following from Klimyk and Schmudgen's book Quantum Groups Page 167:

Proposition For $\frak{g}$ a complex semi-simple Lie algebra, there is an isomorphism $\phi$ of the $h$-adic algebra $U_{h}(\frak{g})$ onto $U({\frak g})[[h]]$ which coincides with the identity map modulo $h$.

Explicitly, for $U_h(\frak{sl}_2)$, the $h$-adic algebra generated by the elements $H,E$, and $F$, such that $$[H,E] = 2E, ~~~~~ [H,F] = -2F, ~~~ [E,F] = (e^{h H} - e^{-hH})/(e^h - e^{-h}).$$ The isomorphism $\phi$ of the algebras is uniquely determined by its action on the generating elements and is given by $$\phi(H) = H' ~~~~~ \phi(F) = F', ~~~~~ \phi(E) = 2\Big(\frac{\cosh h(H'-1) - \cosh 2h\sqrt{C'}}{|H'-1|^2-4C' \sinh^2 h}\Big),$$ where $H', E',F'$ are the generators of $U(\frak{sl}_2)$ satisfying the relations $[H',E'] = 2E'$, $[H',F'] = -2F'$, $[E',F'] = H'$, and $C' = \frac{1}{4}(H'-1)^2 + E'F'$ is the Casimir element of $U_h(\frak{sl}_2)$.

I would like to ask if anyone knows of an explicit description of this isomorphism for the case of $\frak{sl}_3$? The reference for the proof of the proposition is Drinfeld's 1986 ICM talk, or Shnider and Sternberg Quantum Groups Chapter 11.

This post imported from StackExchange MathOverflow at 2014-10-08 13:36 (UTC), posted by SE-user User1298

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.