Drinfeld and Jimbo’s Quantum Enveloping Algebras
In Part I we have investigated at length the quantum enveloping alge-bra of sl(2). In this chapter we give a brief presentation of the algebras Uh(g) associated by Drinfeld [Dri85][Dri87] and Jimbo [Jim85] to the other semisimple Lie algebras g. The algebras Uh(g) provide non-trivial examples of quantum enveloping algebras as defined in XVI.5 as well as examples of isotopy invariants of links. We shall also need Uh(g) in Chapter XIX to state the Drinfeld-Kohno theorem on the monodromy of the Knizhnik-Zamolodchikov systems. Finally, in Section 4 we shall determine an explicit universal R-matrix for the quantum enveloping algebra of sl(2), using the crossed bimodules of IX.5.
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