A torsion theory in the category of cocommutative Hopf algebras

The purpose of this article is to prove that the category of cocommutative Hopf $K$-algebras, over a field $K$ of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie $K$-algebras, respectively

Connected components of compact matrix quantum groups and finiteness conditions

We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in the sense of Wang) connected subgroup by introducing canonical, but possibly transfinite, sequences of subgroups. These sequences have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal and the associated sequence has length 1. We give necessary and sufficient conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A_u(F) is not of Lie type. We discuss an example arising from the compact real form of U_q(sl_2) for q<0.Comment: 43 pages. Changes in the introduction. The relation between our and Wang's notions of central subgroup has been clarifie

A note on split extensions of bialgebras

We prove a universal characterization of Hopf algebras among cocommutative bialgebras over a field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.Comment: Reduced the context to algebraically closed field

The Crystal Duality Principle: from Hopf Algebras to Geometrical Symmetries

We give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras which have some geometrical content. When the ground field has characteristic zero, the first pair is made by a function algebra over a connected Poisson group and a universal enveloping algebra over a Lie bialgebra. In addition, the Poisson group as a variety is an affine space, and the Lie bialgebra as a Lie algebra is graded. Forgetting these last details, the second pair is of the same type. When the Hopf algebra H we start from is already of geometric type the result involves Poisson duality: the first Lie bialgebra associated to H = F[G] is g^* (with g := Lie(G)), and the first Poisson group H = U(g) is of type G^*, i.e. it has g as cotangent Lie bialgebra. If the ground field has positive characteristic, then the same recipes give similar results, but for the fact that the Poisson groups obtained have dimension 0 and height 1, and restricted universal enveloping algebras are obtained. We show how these "geometrical" Hopf algebras are linked to the original one via 1-parameter deformations, and explain how these results follow from quantum group theory. The cases of hyperalgebras and group algebras are examined in some detail (thus recovering some well-known, classical construction), along with some relevant examples.Comment: 37 pages, AMS-TeX file. Minor typoes have been corrected in pages 23, 24, 26, 32 and 36. Final version, to appear in Journal of Algebr

Pushout of quasi-finite and flat group schemes over a Dedekind ring

Let $G$, $G_1$ and $G_2$ be quasi-finite and flat group schemes over a complete discrete valuation ring $R$, $\varphi_1:G\to G_1$ any morphism of $R$-group schemes and $\varphi_2:G\to G_2$ a model map. We construct the pushout $P$ of $G_1$ and $G_2$ over $G$ in the category of $R$-affine group schemes. In particular when $\varphi_1$ is a model map too we show that $P$ is still a model of the generic fibre of $G$. We also provide a short proof for the existence of cokernels and quotients of finite and flat group schemes over any Dedekind ring.Comment: 18 pages, preliminary versio

A semi-abelian extension of a theorem by Takeuchi

We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor-Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context.Comment: 27 page