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

Details Duality for the Jordanian Matrix Quantum Group $GL_{g,h}(2)$ Duality for the Jordanian Matrix Quantum Group $GL_{g,h}(2)$

1997-05-29

arxiv.org/abs/q-alg/9705028v2

J. Phys. A30 (1997) 6769-6781

Mathematics

Quantum Algebra

plain TeX with harvmac, 16 pages, added Appendix implementing the ACC nonlinear map

Scientific paper

10.1088/0305-4470/30/19/016

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 $U_{-h,h}$.

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