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Quantum Weyl group and some of its applications. (English) Zbl 0753.17029
Geometry and physics, Proc. Winter Sch., Srni/Czech. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 233-235 (1991).
[For the entire collection see Zbl 0742.00067.]
The author exposes some of his results on the “quantum Weyl group”, a certain subalgebra of the dual of the algebra of functions on a quantum group, generated by elements satisfying the braid relations. This enables him to find an explicit formula for the universal $$R$$-matrix [see S. Levendorskij and Ya. S. Sojbel’man, J. Geom. Phys. 7, 241-254 (1990; Zbl 0729.17009), and also A. N. Kirillov and N. Reshetikhin, Commun. Math. Phys. 134, 421-431 (1990; Zbl 0723.17014); S. M. Khoroshkin and A. N. Tolstoj, Funkts. Anal. Appl. 26, 69-71 (1992)] and an action of the Hecke algebra on the subspace of vectors of weight 0 in the deformation of the adjoint representation [see also G. Lusztig, Adv. Math. 70, 237-249 (1988; Zbl 0651.17007); J. Algebra 131, 466-475 (1990; Zbl 0698.16007)].

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations