Duality for the Jordanian Matrix Quantum Group Gl G,h (2)

We find the Hopf algebra U g,h dual to the Jordanian matrix quantum group GL g,h (2). As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: U ′ g,h (with three generators) and U (Z) (with one generator). The subalgebra U (Z) is a central Hopf subalgebra of U g,h. The subalgebra U ′ g,h is not a Hopf subalgebra and its coalgebra structure depends on both parameters. We discuss also two one-parameter special cases: g = h and g = −h. The subalgebra U ′ h,h is a Hopf algebra and coincides with the algebra introduced by Ohn as the dual of SL h (2). The subalgebra U ′ −h,h is isomorphic to U (sl(2)) as an algebra but has a nontrivial coalgebra structure and again is not a Hopf subalgebra of U −h,h .