On the quantization of a coherent family of representations at roots of unity

摘要

Given a coherent family of virtual representations of a complex semisimple Lie algebra we associate a coherent family of virtual representations of the corresponding quantum group at roots of unity. The latter family depends on the given family in a precise fashion described below. Let g be a finite dimensional complex semisimple Lie algebra and let U be its universal enveloping algebra. Lusztig considered a certain C[υ, υ~(-1)] algebra U_A, {A=C[υ, υ~(-1)]} which is an A-form of the 'quantum group' U_(A′), {A′-C(υ) the field of fractions of A} ; the latter are some Hopf-algebra deformations of U, defined by Drinfeld and Jimbo generalizing the case of Sγ_2.