# Explicit Descriptions of $q-SO(2)$, and $q-Sp(2)$?

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!

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Have you looked in any books? You should be able to find these in most, if not all, introductory quantum groups books. Off the top of my head, I would check Chapter 9 of the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. – MTS Oct 26 '12 at 16:46
Sometimes $\mathbb{Z}_n$ can serve the role of a quantum $U(1)$ in that it has a quasitriangular structure and is a subalgebra of $U_q SU(2)$ for $q$ the right root of unity. Majid's book has details about that. – Turion May 29 '14 at 14:44

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)

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