The trace formula for braided Hopf algebras

For any n-dimensional Hopf algebra H, there is an important trace formula, due to Larson and Radford [LR1, 2]: Tr(S-2)= epsilon(t)phi(1) = nTr(S-2\Hy), where S is the antipode of H, t a nonzero right integral in H, phi a right integral in H* with phi (t) = 1 and ya specific element in H. It is natural to ask whether there is an analogue of such a formula for braided Hopf algebras, i.e., Hopf algebras in the category of Yetter-Drinfeld modules over a Hopf algebra L. The first equality was recently generalized by Andruskiewitsch and Schneider [AS, Theorem 7.3], using bosonizon process. The main purpose of this paper is to generalize the second equality. We replace the square of the antipode by the map U (see (25) below), and assume in the main result (Theorem 2.6) that L is involutory.