Abstract: In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra $H$ [P] and extend two recent theorems in [E]. We obtain two Radford formulas for the antipode in $H$ and generalize in Section 7 the results on its order in [F]. We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of $H$ is symmetric and unimodular. \item{[P]} Pareigis, B.: On the cohomology of modules over Hopf algebras. J. Algebra 22 (1972), 161-182. \item{[E]} Etingof, P.; Gelaki, S.: On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic. Internat. Math. Res. Notices 16 (1998), 851-864. \item{[F]} Fischman, D.; Montgomery, S.; Schneider, H.-J.: Frobenius extensions of subalgebras of Hopf algebras. Trans. Amer. Math. Soc. 349 (1997), 4857-4895.
Classification (MSC2000): 16W30; 16L60
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