Rational forms for twistings of enveloping algebras of simple Lie algebras (Q1196090)

This paper is concerned with Hopf algebras obtained by ``gauge'' transformations. Let \(A\) be a Hopf algebra, with comultiplication \(\Delta\), antipode \(\mathcal S\) and counit \(\varepsilon\). Let \(F\in A\otimes A\) be an invertible element satisfying \((\varepsilon \otimes 1) F = (1 \otimes \varepsilon) F = 1\). The algebra \(A\) together with \(\Delta^{F} = F\Delta F^{-1}\) and a certain \(\mathcal S^{F}\) is an example of a quasi-Hopf algebra, a notion introduced by \textit{V. G. Drinfel'd} [Leningr. Math. J. 1, No. 6, 1419-1457 (1990), translation from Algebra Anal. 1, No. 6, 114-148 (1989; Zbl 0718.16033)]. If in addition \(F_{23}(1 \otimes \Delta) F =F_{12}(\Delta \otimes 1) F\) then it is a genuine Hopf algebra. Let \(\mathfrak g\) be a complex Lie algebra, \(A\) the \(h\)-adic completion of the universal enveloping algebra of \({\mathfrak g}[[h]]\), \(F = \exp h f\) where \(f\) belongs to a commutative subalgebra \(\mathfrak c\). Then \(F\) satisfies the requirements above (in a topological setting) and hence one obtains new Hopf algebras (which are quantized enveloping algebras for a suitable Lie bialgebra structure on \(\mathfrak g\)) by twisting a universal enveloping algebra. The purpose of this paper is to show that the so-obtained Hopf algebras admit ``rational forms'' when \(\mathfrak g\) is simple and finite dimensional, and \(\mathfrak c\) is, say for simplicity, the Cartan subalgebra of \(\mathfrak g\). These rational forms are defined over a ring \(\mathbb{C}[h, \exp hu_{ij}]\) for some integers \(u_{ij}\). A link with generalized commutative (or braided) algebras is also established.