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Jacobian quasi-bialgebras and quasi-Poisson Lie groups. (English) Zbl 0847.17020

Mathematical aspects of classical field theory, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Seattle/WA (USA) 1991, Contemp. Math 132, 459-489 (1992).
The author considers only the various generalizations of the Lie bialgebras and one of the generalizations of the Poisson Lie groups, the quasi-Poisson Lie groups. In [Leningr. Math. J. 1, 1419-1457 (1990); translation from Algebra Anal. 1, No. 6, 114-148 (1989; Zbl 0718.16033)], V. G. Drinfeld introduced and studied the associative quasi-bialgebras, which he called quasi-bialgebras, and the associative quasi-Hopf-algebras, which he called quasi-Hopf algebras. S. Majid has treated the coassociative quasi-Hopf algebras, calling them dual quasi-Hopf algebras [Sh. Majid, 134, 219-232 (1992; Zbl 0788.17012)]. The case of the Poisson quasi-groups has not yet been fully investigated. Most of the results in this lecture were announced in [Y. Kosmann-Schwarzbach, C. R. Acad. Sci., Paris, Sér I 312, 123-126 (1991; Zbl 0726.17029); ibid., 233-236 (1991; Zbl 0726.17030); ibid., 391-394 (1991; Zbl 0712.22012)]. A survey of the category-theoretic approach to quasi-quantum groups as well as a short summary of the theory of quasi-Poisson Lie groups are available in [Y. Kosmann-Schwarzbach, From “quantum groups” to “quasi-quantum groups”, in Symmetries in science V, Algebraic systems, their representations, realizations and physical applications, B. Gruber, L. C. Bieden harn and H. D. Doebner eds., 369-393 (1991)].
For the entire collection see [Zbl 0755.00014].

MSC:

17B66 Lie algebras of vector fields and related (super) algebras
17B56 Cohomology of Lie (super)algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
58H05 Pseudogroups and differentiable groupoids
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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