We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group BQ(k, E(n)) of E(n) that are isomorphic. For a triangular structure R on E(n) we prove that the subgroup BM (k, E(n), R) of BQ(k, E(n)) arising from R is isomorphic to a direct product of BW (k), the BrauerWall group of the ground field k, and Sym n (k), the group of n × n symmetric matrices under addition. For a general quasi-triangular structure R ′ on E(n) we construct a split short exact sequence having BM (k, E(n), R ′ ) as a middle term and as kernel a central extension of the group of symmetric matrices of order r < n (r depending on R ′ ). We finally describe how the image of the Hopf automorphism group inside BQ(k, E(n)) acts on Sym n (k).

Abstract. We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).

Abstract. We study actions of semisimple Hopf algebras H on Weyl algebras A over an algebraically closed field of characteristic zero. We show that the action of H on A must factor through a group action; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.

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In this paper we extend the theory of serial and uniserial ÿnite dimensional algebras to coalgebras of arbitrary dimension. Nakayama-Skorniakov Theorems are proved in this new setting and the structure of such coalgebras is determined up to Morita-Takeuchi equivalences. Our main structure theorem asserts that over an algebraically closed ÿeld k the basic coalgebra of a serial indecomposable coalgebra is a subcoalgebra of a path coalgebra k where the quiver is either a cycle or a chain (ÿnite or inÿnite). In the uniserial case, is either a single point or a loop. For cocommutative coalgebras, an explicit description is given, serial coalgebras are uniserial and these are isomorphic to a direct sum of subcoalgebras of the divided power coalgebra.

We show that there is a family of complex semisimple Hopf algebras that do not admit a Hopf order over any number ring. They are Drinfel'd twists of certain group algebras. The twist contains a scalar fraction which makes impossible the definability of such Hopf algebras over number rings. We also prove that a complex semisimple Hopf algebra satisfies Kaplansky's sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.Given a twist J for H a new Hopf algebra H J , called Drinfel'd twist of H, can be constructed as follows: H J = H as an algebra, the counit is that of H, and the new comultiplication and antipode are:Here U J := J (1) S(J (2) ) and, if we write J −1 = J −(1) ⊗ J −(2) , its inverse is U −1 J = S(J −(1) )J −(2) . We next describe the particular example of twist for the group algebra that we will use. This construction was used several times in the past, see for example, [6], [8] or [18] and the references therein. Let M be a finite abelian group and M its group of characters. Consider the group algebra KM and its dual Hopf algebra (KM ) * . We identify (KM ) * and K M as Hopf algebras. Assume that char K ∤ |M |. There is an isomorphism of Hopf algebras (KM ) * ≃ KM induced by an isomorphism of groups M ≃ M . Let {ϑ m } m∈M ⊂ (KM ) * be the dual basis of {m} m∈M . If ω : M × M → K × is a normalized 2-cocycle, then J = m,m ′ ∈M ω(m, m ′ )ϑ m ⊗ ϑ m ′ ∈ (KM ) * ⊗ (KM ) *

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We show that the Brauer group BM(k, H_v, R_{s,beta}) of the quasitriangular Hopf algebra (H_v, R_{s,beta}) is a direct product of the additive group of the field k and the classical Brauer group B_{\theta_s}(k, Z_{2v}) associated to the bicharacter theta_s on Z_{2v}, defined by theta_s(x, y) = omega^(swy), with omega a 2vth root of unity