Quasi-Lie bialgebras and Lie loops (Q1043697)

Lie-Poisson groups and Lie bialgebras are dual notions. Quasi-Poisson Lie groups were studied by \textit{Y. Kosmann-Schwarzbach} [C. R. Acad. Sci., Paris, Sér. I 312, No. 5, 391--394 (1991; Zbl 0712.22012)] as classical limits of quasi-Hopf algebras, whose infinitesimals objects were called quasi-Lie bialgebras by \textit{V. G. Drinfeld} [Leningr. Math. J. 1, No. 6, 1419--1457 (1990); translation from Algebra Anal. 1, No. 6, 114--148 (1989; Zbl 0718.16033)], but translated in French as Lie quasi-bialgebras. However, quasi-Poisson Lie groups and Lie quasi-bialgebras are not dual notions. The dual notion of a Lie quasi-bialgebra is called in the paper under review a quasi-Lie bialgebra (not to be confused with Drinfeld's use of the term). The author constructs the dual object of a quasi-Poisson Lie group, i.e., a geometric object generalizing Poisson-Lie groups, whose tangent space at a distinguished element has the structure of a quasi-Lie bialgebra. Since the terminology is just as confusing in French as in English, we try to indicate what these dual objects are. A quasi-Lie bialgebra is a finite-dimensional real or complex vector space \(F\) with a Lie algebra structure \(m\) and a Lie coalgebra structure \(c\) with \(c\) an \(m\)-derivation, as in a Lie bialgebra, but also involves a 3-chain in the triple wedge product of \(F^*\) satisfying coboundary and symmetry type conditions. In this case, \(F^*\) is a Lie quasi-bialgebra (see the reference to Kosmann-Schwarzback above). A quasi-Poisson Lie quasi-group involves a right alternative Lie loop \(B\), a Lie group \(G\) acting on the right of \(B\), a bivector field vanishing on the identity element of \(B\), an analytic map \(f\) of \(B \times B\) to \(G\), an \(f\)-twisted left action of \(B\) on \(G\), and a real non-degenerate invariant bilinear form on the product of the tangent spaces of \(G\) and \(B\) at their identity elements, which satisfy four axioms. The main theorem of the paper says that for a local quasi-Poisson Lie quasi-group, the tangent space at the identity element has the structure of a quasi-Lie bialgebra, and conversely, for every real quasi-Lie bialgebra, there corresponds a unique local quasi-Poisson Lie quasi-group whose quasi-Lie bialgebra tangent space coincides with the given quasi-Lie bialgebra. The discussion involves Akivis algebras, quasi-double Lie groups and quasi-double Lie algebras. The latter two generalize double Lie groups and double Lie algebras.