Reconstruction of Hidden Symmetries

Representations of a group $G$ in vector spaces over a field $K$ form a category. One can reconstruct the given group $G$ from its representations to vector spaces as the full group of monoidal automorphisms of the underlying functor. This is a special example of Tannaka-Krein theory. This theory was used in recent years to reconstruct quantum groups (quasitriangular Hopf algebras) in the study of algebraic quantum field theory and other applications. We show that a similar study of representations in spaces with additional structure (super vector spaces, graded vector spaces, comodules, braided monoidal categories) produces additional symmetries, called ``hidden symmetries''. More generally, reconstructed quantum groups tend to decompose into a smash product of the given quantum group and a quantum group of ``hidden'' symmetries of the base category.Comment: 42 pages, amslatex, figures generated with bezier.sty, replaced to facilitate mailin

On the Cohomology of Modules over Hopf Algebras

AbstractLet R be a commutative ring. Define an FH-algebra H to be a Hopf algebra and a Frobenius algebra over R with a Frobenius homomorphism ψ such that ∑(h) h(1) ψ(h(2)) = ψ(h) · 1 for all h ϵ H. This is essentially the same as to consider finitely generated projective Hopf algebras with antipode. For modules over FH-algebras we develop a cohomology theory which is a generalization of the cohomology of finite groups. It generalizes also the cohomology of finite-dimensional restricted Lie algebras. In particular the following results are shown. The complete homology can be described in terms of the complete cohomology. There is a cup-product for the complete cohomology and some of the theorems for periodic cohomology of finite groups can be generalized. We also prove a duality theorem which expresses the cohomology of the “dual” of an H-module as the “dual” of the cohomology of the module. The last section provides techniques to describe under certain conditions the cohomology of H by the cohomology of sub- and quotient-algebras of H. In particular we have a generalization of the Hochschild-Serre spectral sequence for the cohomology of groups

When Hopf Algebras are Frobenius Algebras

AbstractR. Larson and M. Sweedler recently proved that for free finitely generated Hopf algebras H over a principal ideal domain R the following are equivalent: (a) H has an antipode and (b) H has a nonsingular left integral. In this paper I give a generalization of this result which needs only a minor restriction, which, for example, always holds if pic(R) = 0 for the base ring R. A finitely generated projective Hopf algebra H over R has an antipode if and only if H is a Frobenius algebra with a Frobenius homomorphism ψ such that Σ h(1) ψ(h(2)) = ψ(h) · 1 for all h ϵ H. We also show that the antipode is bijective and that the ideal of left integrals is a free rank 1, R-direct summand of H

Forms of Hopf Algebras and Galois Theory

The theory of Hopf algebras is closely connected with various applications, in particular to algebraic and formal groups. Although the rst occurence of Hopf algebras was in algebraic topology, they are now found in areas as remote as combinatorics and analysis. Their structure has been studied in great detail and many of their properties are well understood. We are interested in a systematic treatment of Hopf algebras with the techniques of forms and descent. The rst three paragraphs of this paper give a survey of the present state of the theory of forms of Hopf algebras and of Hopf Galois theory especially for separable extensions. It includes many illustrating examples some of which cannot be found in detail in the literature. The last two paragraphs are devoted to some new or partial results on the same eld. There we formulate some of the open questions which should be interesting objects for further study. We assume throughout most of the paper that k is a base eld and do not touch upon the recent beautiful results of Hopf Galois theory for rings of integers in algebraic number elds as developed in [C1]