Classification of bicovariant differential calculi on the Jordanian quantum groups GL_{g,h}(2) and SL_{h}(2) and quantum Lie algebras

Abstract

We classify all 4-dimensional first order bicovariant calculi on the Jordanian quantum group GL_{h,g}(2) and all 3-dimensional first order bicovariant calculi on the Jordanian quantum group SL_{h}(2). In both cases we assume that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators. It is found that there are 3 1-parameter families of 4-dimensional bicovariant first order calculi on GL_{h,g}(2) and that there is a single, unique, 3-dimensional bicovariant calculus on SL_{h}(2). This 3-dimensional calculus may be obtained through a classical-like reduction from any one of the three families of 4-dimensional calculi on GL_{h,g}(2). Details of the higher order calculi and also the quantum Lie algebras are presented for all calculi. The quantum Lie algebra obtained from the bicovariant calculus on SL_{h}(2) is shown to be isomorphic to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian universal enveloping algebra U_{h}(sl(2)) and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module. We also obtain the quantum Killing form for this quantum Lie algebra.

33 pages, AMSLaTeX, misleading remark removed

References in corpus (6)

• Noncommutative Geometry of the h-deformed Quantum Plane
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• Irreducible decomposition for tensor prodect representations of Jordanian quantum algebras
• Representations and Clebsch-Gordan coefficients for the Jordanian quantum algebra U_h(sl(2))
• The structure of quantum Lie algebras for the classical series B_l, C_l and D_l
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Cited by in corpus (1)

• Classification of Left-Covariant Differential Calculi on the Quantum Group $\SLq 2$