# Does There Exists a General Quantum Casimir Extending the $U_q({\mathfrak sl}_2$ Case?

As is well known (see Kassel), when $q$ is not a root of unity, the centre or the quantum enveloping algebra $U_q({\mathfrak sl}_2)$ of ${\mathfrak sl}_2$ is generated by the element $$C_q = EF + \frac{q^{-1}K+qK^{-1}}{(q-q^{-1})^2}.$$ The element is called the quantum Casimir. My questions are as follows:

(i) Does this situation extend to the general setting of $U_q({\mathfrak sl}_N)$?

(ii) If it does, is there a general formula for $C_q$?

(iii) How would this formula relate to the usual formula for the classical Casimir? (The uasual formula I refer to is $\sum X^iX_i$, for some basis $X_i$ and its dual $X^i$, see wikipedia for details.)

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for the third part, set K=q^h and take the limit as q goes to 1. – Peter McNamara Nov 19 '10 at 18:42
@Peter McNamara: here you mean "$q^H$", where $H$ is the third element of usual $\mathfrak{sl}(2)$ (as opposed to $h = \log q$). – Theo Johnson-Freyd Nov 20 '10 at 2:02

The centers of the Drinfeld-Jimbo quantum groups $U_q(\mathfrak{g})$ are well-understood and quite analogous to the classical case. See the book by Klimyk and Schmüdgen, Section 6.3, where in particular the quantum Casimirs are constructed.
Does the object they construct there reduce to the ordinary Casimir in the $q=1$ case? – Dyke Acland Nov 19 '10 at 19:07
What do you mean by "reduce to"? They have properties analogous to the usual Casimirs. However, looking at their eigenvalues on finite-dimensional modules, it might be necessary to add some constant (rational function of $q$) before you will get the exact $q$-numbers of the usual eigenvalues. I guess it depends on what you need them for. If you would like to take a formal limit you need to look at the $h$-adic versions of these quantum groups. There I am sure there exist proper direct analogues. – Jonas Hartwig Nov 19 '10 at 19:30
I'm just trying to understand these objects. The way I do this with a new quantum object is to see exactly what it is when $q=1$. – Dyke Acland Nov 19 '10 at 19:45
Sure, morally $q=1$ should correspond to classical things. But for example in the quantum casimir you mention one cannot just put $q=1$... – Jonas Hartwig Nov 20 '10 at 3:49