A classical result of Larson and Sweedler says that a finite dimensional Hopf algebra over a field has invertible antipode. Does this result extend to the setting of Hopf algebras in braided categories? In other words, if a Hopf algebra in a braided category is dualizable, is its antipode necessarily invertible? (Equivalently, if a dualizable bialgebra has invertible fusion operator, is its opfusion operator invertible?) If this result is not true, is there an "easy" example where it fails? Of course, if it fails, it should do so in the free braided category equipped with a Hopf algebra, but it seems difficult to see that things aren't invertible using string diagrams.
Yes, they do. See Theorem 4.1 in "Finite Hopf algebra in braided tensor categories" M. Takeuchi, Journal Pure and Applid Algebra 138 (1999) 59-82