# Outer automorphism for $U_q(\mathfrak{su}(2|2))$

+ 2 like - 0 dislike
470 views

It is known that Lie superalgebra $\mathfrak{su}(2|2)$ (and only this one, not arbitrary $\mathfrak{su}(n|n)$) has the nontrivial central extension which forms an $\mathfrak{sl}_2$ triplet, let's call it $(\mathfrak{C},\mathfrak{P},\mathfrak{K})$. The easiest way to see this is to start from the superalgebra $\mathfrak{d}(2,1,\alpha)$ and take $\alpha=0$. An $\mathfrak{sl}_2$ automorphism allows one to rotate the vector $(\mathfrak{C},\mathfrak{P},\mathfrak{K})$ and transform it to e.g. $(\mathfrak{C}',0,0)$ for some $\mathfrak{C}'$. It turns out that representation theory of the algebra with the central extension $(\mathfrak{C}',0,0)$ (when two elements vanish) is much easier than the one with a full $(\mathfrak{C},\mathfrak{P},\mathfrak{K})$.

Now let us consider quantum deformation of of this algebra - $U_q(\mathfrak{su}(2|2))$. The question is how to generalize the outer automorphism to this case if one exists at all. I need this to build up a representation theory for the above superalgebra.

I understand that the question is technical, it's hard to realize its complexity without doing any explicit calculations, but, just in case, if anybody thought about something related, please let me know.

This post imported from StackExchange MathOverflow at 2014-07-29 11:45 (UCT), posted by SE-user Peter

 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:$\varnothing\hbar$ysicsOverflowThen 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.