On quantum group SL_q(2)
  We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group.   Based on this observation we develop a general method of constructing quantum groups with similar property. We also describe this method in the language of quantized universal enveloping algebras, which is another common method of studying quantum groups.   We carry our method in detail for root systems of type SL(2); as a byproduct we find a new series of quantum groups - metaplectic groups of SL(2)-type. Representations of these groups can provide interesting examples of bimodule categories over monoidal category of representations of SL_q(2). 
Joseph Bernstein, Tanya Khovanova