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We classify all 4-dimensional first order bicovariant calculi on theJordanian quantum group GL_{h,g}(2) and all 3-dimensional first orderbicovariant calculi on the Jordanian quantum group SL_{h}(2). In both cases weassume that the bicovariant bimodules are generated as left modules by thedifferentials of the quantum group generators. It is found that there are 31-parameter families of 4-dimensional bicovariant first order calculi onGL_{h,g}(2) and that there is a single, unique, 3-dimensional bicovariantcalculus on SL_{h}(2). This 3-dimensional calculus may be obtained through aclassical-like reduction from any one of the three families of 4-dimensionalcalculi on GL_{h,g}(2). Details of the higher order calculi and also thequantum Lie algebras are presented for all calculi. The quantum Lie algebraobtained from the bicovariant calculus on SL_{h}(2) is shown to be isomorphicto the quantum Lie algebra we obtain as an ad-submodule within the Jordanianuniversal enveloping algebra U_{h}(sl(2)) and also through a consideration ofthe decomposition of the tensor product of two copies of the deformed adjointmodule. We also obtain the quantum Killing form for this quantum Lie algebra.Comment: 33 pages, AMSLaTeX, misleading remark remove

Classification of bicovariant differential calculi on the Jordanian quantum groups and and quantum Lie algebras