Classification of bicovariant differential calculi on the Jordanian quantum groups GLh,g (2) and SLh (2) and quantum Lie algebras

Assuming that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators we classify all four-dimensional first-order bicovariant calculi on the Jordanian quantum group GL(h.g)(2) and all three-dimensional first-order bicovariant calculi on the Jordanian quantum group SLh(2). It is found that there are three one-parameter families of four-dimensional bicovariant first-order calculi on GLh.g(2) and that there is a single, unique, three-dimensional bicovariant calculus on SLh(2). This three-dimensional calculus may be obtained through a classical-like reduction from any one of the three families of four-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 SLh (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)(C)) 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.