S. Zakrzewski constructed compact or locally compact quantum groups with deformation parapmeter $q=-1$ for subgroups of $GL(2;C)$, e.g. $SU_{-1}(2)$, $SU_{-1}(1,1)$, $SL_{-1}(2,R)$, as algebras of functions on the classical (=undeformed) group with values in $2\times 2$ matrices, see

S. Zakrzewski, Matrix pseudogroups associated with anti-commutative plane, Letters in Mathematical Physics, April 1991, Volume 21, Issue 4, pp 309-321.

He writes that it is likely that similar constructions are possible in higher dimensions.

T. Banica and J. Bichon constructed a similar matrix model with $4\times 4$ matrices for $S_4^+\cong SO_{-1}(3)$ using the theory of cocycle twists, see

T. Banica and J. Bichon, Quantum groups acting on 4 points, Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2009, Issue 626, Pages 75–114, January 2009.

The method of cocycle twists can of course be applied also to other groups. I expect that $q$-deformations of other simple Lie groups with $q=-1$ can be constructed in the same way. Does anybody know if work in this direction has already been carried out and published?

Many thanks in advance!

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    $\begingroup$ Indeed, that's the case, e.g., for Banica and Bichon's $SO_{-1}(3)$. I think one could start from the list in Korogodski and Soibelman, but I'd also be happy with additional (new?) examples. $\endgroup$
    – Uwe Franz
    Jan 8, 2013 at 4:00

1 Answer 1


In their new paper "Quantum subgroups of the compact quantum group SU_{-1}(3)", (see http://arxiv.org/abs/1306.6244), Julien Bichon and Robert Yuncken give a positive answer to this question for $SU_{-1}(2m+1)$ and $U_{-1}(2m+1)$.


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