– We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p > 17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius–Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. ...

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations an...

Let R be the valuation ring ofK, a finite extension of Qp containing a primitive pth root of unity, and let G be an elementary abelian p-group of order p, with dual group Ĝ. We construct a new family of triangular Hopf orders over R in KG, a proper subfamily whose duals are also triangular, and a proper subfamily of that family whose construction extends the truncated exponential construction o...

The aim of this section is to define some generalization of the notion of formal group. More precisely, we consider the analog of formal groups with coefficients belonging to a Hopf algebra. We also study some example of a formal group over a Hopf algebra, which generalizes the formal group of geometric cobordisms. Recently some important connections between the Landweber-Novikov algebra and th...

We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf-Pasquier-Roche quasi-Hopf algebra D(Σ), for some finite group Σ and some ω ∈ Z(Σ, k×). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Σ are group theoretical. We also describe their Drinfeld double as ...

Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds. They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. In this paper we define a state sum invariant of triangulated 4-manifolds usi...

The goal of this paper is to give a new method of constructing finite-dimensional semisimple triangular Hopf algebras, including minimal ones which are non-trivial (i.e. not group algebras). The paper shows that such Hopf algebras are quite abundant. It also discovers an unexpected connection of such Hopf algebras with bijective 1-cocycles on finite groups and set-theoretical solutions of the q...

A C∞-Hopf algebra is a C∞-algebra which is also a convenient Hopf algebra with respect to the structure induced by the evaluations of smooth functions. We characterize those C∞-Hopf algebras which are given by the algebra C∞(G) of smooth functions on some compact Lie group G, thus obtaining an anti-isomorphism of the category of compact Lie groups with a subcategory of convenient Hopf algebras.

We show that if a finite dimensional Hopf algebra over C has a basis such that all the structure constants are non-negative, then the Hopf algebra must be given by a finite group G and a factorization G = G+G− into two subgroups. We also show that Hopf algebras in the category of finite sets with correspondences as morphisms are classified in the similar way. Our results can be used to explain ...