When ever I hear noncommutative geometers talking about quantum groups, it is usually $q-SU(2)$ that they are discussing. As a result there are many good and explicit generator and relation presentations of this Hopf algebra. For an easy example take this other M.O. question. I am curious to see what the simple examples of the other quantum groups series are. More specifically, could anyone give me a generator and relation description of the Hopf algebras $$ q-SO(2), ~~~~~~~~~~~~~~~~~~~ q-Sp(2)? $$ I know that somehow these are derivable from the some quantized enveloping algebra dual quantum groups, but that's a little too difficult for a ''classical'' geometer like me!
If I am not mistaken, $SO(2)\approx U(1)$ has no nontrivial quantum deformation, but $SO(3)$ does; this is explicitly constructed in:
Quantum SO(3) groups, P.M. Soltan (2008)
For q-Sp(2), see Section 3.2 of
Noncommutative families of instantons, G. Landi et al. (2008).
More general references on quantum classical groups:
Quantum deformation of classical groups, T. Hayashi (1992)
Quantum symmetric spaces and related q-orthogonal polynomials, M. Noumi and T. Sugitani (1995)
Orthogonal and symplectic quantum matrix algebras and Cayley-Hamilton theorem for them, O. Ogievetsky and P. Pyatov (2005)