LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗
 B
 A →A ⊗H the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.

Principal homogeneous spaces for arbitrary Hopf algebras

This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras A by first determining the graded Hopf algebra grA associated to the coradical filtration of A. The A0-coinvariants elements form a braided Hopf algebra R in the category of Yetter–Drinfeld modules over the coradical A0 = |Γ, Γ the group of group-like elements of A, and gr A ' R#A0. We call the braiding of the primitive elements of R the infinitesimal braiding of A. If this braiding is of Cartan type [AS2], then it is often possible to determine R, to show that R is generated as an algebra by its primitive elements and finally to compute all deformations or liftings, that is pointed Hopf algebras such that gr A ' R#|Γ. In the last chapter, as a concrete illustration of the method, we describe explicitly all finite-dimensional pointed Hopf algebras A with abelian group of group-likes G(A) and infinitesimal braiding of type An (up to some exceptional cases). In other words, we compute all the liftings of type An; this result is our main new contribution in this paper.

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Pointed Hopf Algebras

(1986). Cleft comodule algebras for a bialgebra. Communications in Algebra: Vol. 14, No. 5, pp. 801-817.

Cleft comodule algebras for a bialgebra

(1996). Hopf bigalois extensions. Communications in Algebra: Vol. 24, No. 12, pp. 3797-3825.

Hopf bigalois extensions

Normal basis and transitivity of crossed products for Hopf algebras

Multiplication alteration by two-cocycles - the quantum version

(1993). Braided bialgebras and quadratic blalgebras. Communications in Algebra: Vol. 21, No. 5, pp. 1731-1749.

Braided bialgebras and quadratic blalgebras

On the structure of relative hopf modules

Equivalent crossed products for a hopf algebra

(1992). Genaralization fo cleft comodule algabras. Communications in Algebra: Vol. 20, No. 12, pp. 3703-3721.

Yukio Doi

Papers

Multiplication alteration by two-cocycles - the quantum version

Braided bialgebras and quadratic blalgebras

On the structure of relative hopf modules

Equivalent crossed products for a hopf algebra

Genaralization fo cleft comodule algabras