Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif. (Quantized enveloping algebras, compact quantum matrix groups and noncommutative differential calculus) (Q2640687)

The article is in line with earlier investigations of the author [C. R. Acad. Sci., Paris, Sér. I 304, 323-326 (1987; Zbl 0617.16005)] treating the relationships between the two well-known approaches to the abstract notion of a quantum group, one due to \textit{V. G. Drinfel'd} [Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)] and \textit{M. Jimbo} [Lett. Math. Phys. 10, 63-69 (1985; Zbl 0587.17004)], as deformations of universal enveloping algebras, and the other approach due to \textit{S. L. Woronowicz} [Publ. Res. Inst. Math. Sci. 23, No.1, 117-181 (1987; Zbl 0676.46050)], where the basic notion is that of a compact matrix pseudogroup. The author succeeds in assigning compact matrix pseudogroups (of the types B, C, and D) to the corresponding q-groups in the sense of Drinfel'd-Jimbo. The key role is played by the Tannaka-Krein type duality. Also, a first order differential calculus (in the sense of Woronowicz) is built over the resulting compact matrix pseudogroups.