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We extend the main result of Skryabin in [9] to Yetter-Drinfeld Hopf algebras over an arbitrary Hopf algebraHwith bijective antipode. [ABSTRACT FROM AUTHOR]

Abstract: The key part of the definition of a Hopf-Galois extension over the Hopf algebra is bijectivity of a canonical map . We develop criteria under which surjectivity of (which is usually much easier to verify) is sufficient, and we investigate the consequences for the structure of as a -module and -comodule. In particular, we prove equivariant projectivity of extensions in several important cases. We study these questions for generalizations of -Galois extensions like -Galois extensions for a quotient coalgebra and one-sided module of a Hopf algebra , and coalgebra Galois extensions. [Copyright &y& Elsevier]

The trace of powers of the square of the antipode s2 of a finite-dimensional Hopf algebra A over a field k is studied. It is shown in many cases that the trace function vanishes on s2m when s2m ≠ 1A. Finer properties of the antipode are related to this phenomenon. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the representation theory of a large class of pointed Hopf algebras, extending results of Lusztig and others. We classify all simple modules in a suitable category and determine the weight multiplicities; we establish a complete reducibility theorem in this category. [Copyright &y& Elsevier]