Duality for the Jordanian Matrix Quantum Group $G{L}_{g,h}(2)$

Abstract

We find the Hopf algebra $U_{g,h}$ dual to the Jordanian matrix quantum group $G{L}_{g,h}(2)$. As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: $U_{g,h}^{\prime }$ (with three generators) and $U(Z)$ (with one generator). The subalgebra $U(Z)$ is a central Hopf subalgebra of $U_{g,h}$. The subalgebra $U_{g,h}^{\prime }$ 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 $U_{h,h}^{\prime }$ is a Hopf algebra and coincides with the algebra introduced by Ohn as the dual of $S{L}_h(2)$. The subalgebra $U_{-h,h}^{\prime }$ is isomorphic to $U(sl(2))$ as an algebra but has a nontrivial coalgebra structure and again is not a Hopf subalgebra of $U_{-h,h}$.

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