The order of the antipode of a Hopf algebra

Abstract

The coalgebra is cocommutative if t8 = 3, where t: CO C-->C@ C is defined by t(c 0 d) = d 0c. A Hopf algebra over K is an associative algebra H with identity, together with identity-preserving algebra morphisms 3: H->H0H and e: H->K which give the underlying vector space a coalgebra structure. If H is an associative algebra, we will denote by A: HOH->H the map defined by 1.(h0k) =hk, and by 7: K-->H the map defined by 'q(a) =al. An antipode for the Hopf algebra H is a map y: H->H satisfying