On cotriangular Hopf algebras

In an earlier paper, we proved that any triangular semisimple Hopf algebra over an algebraically closed field k of characteristic 0 is obtained from t he group algebra k[G] of a finite group G, by twisting its comultiplication by a twist in the sense of Drinfeld. In this paper, we generalize this res ult to not necessarily finite-dimensional cotriangular Hopf algebras. Namel y, our main result says that a cotriangular Hopf algebra A over k is obtain ed from a function algebra of a proalgebraic group by twisting its multipli cation by a Hopf 2-cocycle, and possibly changing its R-form by a central g rouplike element of A* of order less than or equal to 2, if and only if the trace of the squared antipode on any finite-dimensional subcoalgebra of A is the dimension of this subcoalgebra. The generalization, like the origina l theorem, is proved using Deligne's theorem on Tannakian categories. We th en give examples of twisted function algebras, and in particular, show that in the infinite-dimensional case, the squared antipode may not equal the i dentity. On the other hand, we show that in all of our examples, the square d antipode is unipotent, and conjecture it to be the case for any twisted f unction algebra. We prove this conjecture in a large number of special case s, using the quantization theory of the first author and D. Kazhdan.