# Is there a good differential calculus for quantum SU(3)?

For quantum $SU(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $SU(2)$ by $a,b,c,d$, then the ideal of ker($\epsilon)$ corresponding to this calculus is $$< a+ q^2d - (1+q^2),b^2,c^2,bc,(a-1)b,(d-1)c>.$$ This calculus can be shown to generalise the classical calculus on $SU(2)$ when $q=1$. Does anyone know of a (good) calculus (and its ideal) for quantum $SU(3)$?

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By "good" you mean bicovariant? –  javier Dec 15 '09 at 1:55
I suppose I mean one that is free as a module over SUq(2) and reduces to the classical calculus when q=1. –  Abtan Massini Dec 15 '09 at 3:05
.... and at least left or right covariant. –  Abtan Massini Dec 15 '09 at 3:06

## 1 Answer

I do not know whether it fits all of your requirements, but at least going by the abstract, some version of Woronowicz' result was generalized to all of the quantum groups of classical type in Differential calculus on quantized simple Lie groups, by Branislav Jurco.

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Thanks for the link. The problem is that for $SU(3)$, indeed for all $SU(n)$, the resulting calculus is not of classical dimension. In the case of $SU(2)$ we get Woronowicz's 4-dimensional calculus, not his 3-dimensional calculus. I suppose I should have included the stipulation in the question that the calculus have classical dimension. –  Abtan Massini Mar 11 '10 at 20:50