Pre-torsors and Galois comodules over mixed distributive laws

We study comodule functors for comonads arising from mixed distributive laws. Their Galois property is reformulated in terms of a (so-called) regular arrow in Street's bicategory of comonads. Between categories possessing equalizers, we introduce the notion of a regular adjunction. An equivalence is proven between the category of pre-torsors over two regular adjunctions $(N_A,R_A)$ and $(N_B,R_B)$ on one hand, and the category of regular comonad arrows $(R_A,\xi)$ from some equalizer preserving comonad ${\mathbb C}$ to $N_BR_B$ on the other. This generalizes a known relationship between pre-torsors over equal commutative rings and Galois objects of coalgebras.Developing a bi-Galois theory of comonads, we show that a pre-torsor over regular adjunctions determines also a second (equalizer preserving) comonad ${\mathbb D}$ and a co-regular comonad arrow from ${\mathbb D}$ to $N_A R_A$, such that the comodule categories of ${\mathbb C}$ and ${\mathbb D}$ are equivalent.Comment: 34 pages LaTeX file. v2: a few typos correcte

Weak Projections onto a Braided Hopf Algebra

We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.12. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra $A$ is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that $A$ is a double cross product, or biproduct in Madjid's terminology. The last result is based on a universal property of double cross products which, by Theorem 2.15, works in braided monoidal categories. We also investigate the situation when the right action of the associated matched pair is trivial

Generalized diagonal crossed products and smash products for quasi-Hopf algebras. Applications

In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may be applied to H^*-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the "generating matrix" formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products.Comment: 41 pages, no figure

On the trace of the antipode and higher indicators

We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dim H. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator.Comment: additional references, fixed some typos, minor additions including a questions and some remark

Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups

We classify Lagrangian subcategories of the representation category of a twisted quantum double of a finite group. In view of results of 0704.0195v2 this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of the representation category of a twisted quantum double of a finite group G and module categories over the category of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld's characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.Comment: 26 pages; several comments and references adde

Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra

We rederive a popular nonsemisimple fusion algebra in the braided context, from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter-Drinfeld modules, this strongly suggests that the relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in the (p,1) logarithmic models of conformal field theory. For this, the category of Yetter-Drinfeld modules is to be regarded as an \textit{entwined} category (the one with monodromy, but not with braiding).Comment: 36 pages, amsart++, times, xy. V3: references added, an unnecessary assumption removed, plus some minor change

KAT Ligation for Rapid and Facile Covalent Attachment of Biomolecules to Surfaces

The efficient and bioorthogonal chemical ligation reaction between potassium acyltrifluoroborates (KATs) and hydroxylamines (HAs) was used for the surface functionalization of a self-assembled monolayer (SAM) with biomolecules. An alkane thioether molecule with one terminal KAT group (S-KAT) was synthesized and adsorbed onto a gold surface, placing a KAT group on the top of the monolayer (KAT-SAM). As an initial test case, an aqueous solution of a hydroxylamine (HA) derivative of poly(ethylene glycol) (PEG) (HA-PEG) was added to this KAT-SAM at room temperature to perform the surface KAT ligation. Quartz crystal microbalance with dissipation (QCM-D) monitoring confirmed the rapid attachment of the PEG moiety onto the SAM. By surface characterization methods such as contact angle and ellipsometry, the attachment of PEG layer was confirmed, and covalent amide-bond formation was established by X-ray photoelectron spectroscopy (XPS). In a proof-of-concept study, the applicability of this surface KAT ligation for the attachment of biomolecules to surfaces was tested using a model protein, green fluorescent protein (GFP). A GFP was chemically modified with an HA linker to synthesize HA-GFP and added to the KAT-SAM under aqueous dilute conditions. A rapid attachment of the GFP on the surface was observed in real time by QCM-D. Despite the fact that such biomolecules have a variety of unprotected functional groups within their structures, the surface KAT ligation proceeded rapidly in a chemoselective manner. Our results demonstrate the versatility of the KAT ligation for the covalent attachment of a variety of water-soluble molecules onto SAM surfaces under dilute and biocompatible conditions to form stable, natural amide bonds

Rigidity and defect actions in Landau-Ginzburg models

Studying two-dimensional field theories in the presence of defect lines naturally gives rise to monoidal categories: their objects are the different (topological) defect conditions, their morphisms are junction fields, and their tensor product describes the fusion of defects. These categories should be equipped with a duality operation corresponding to reversing the orientation of the defect line, providing a rigid and pivotal structure. We make this structure explicit in topological Landau-Ginzburg models with potential x^d, where defects are described by matrix factorisations of x^d-y^d. The duality allows to compute an action of defects on bulk fields, which we compare to the corresponding N=2 conformal field theories. We find that the two actions differ by phases.Comment: 53 pages; v2: clarified exposition of pivotal structures, corrected proof of theorem 2.13, added remark 3.9; version to appear in CM

Identification of baryon resonances in central heavy-ion collisions at energies between 1 and 2 AGeV

The mass distributions of baryon resonances populated in near-central collisions of Au on Au and Ni on Ni are deduced by defolding the $p_t$ spectra of charged pions by a method which does not depend on a specific resonance shape. In addition the mass distributions of resonances are obtained from the invariant masses of $(p, \pi^{\pm})$ pairs. With both methods the deduced mass distributions are shifted by an average value of -60 MeV/c$^2$ relative to the mass distribution of the free $\Delta(1232)$ resonance, the distributions descent almost exponentially towards mass values of 2000 MeV/c^2. The observed differences between $(p, \pi^-)$ and $(p, \pi^+)$ pairs indicate a contribution of isospin $I = 1/2$ resonances. The attempt to consistently describe the deduced mass distributions and the reconstructed kinetic energy spectra of the resonances leads to new insights about the freeze out conditions, i.e. to rather low temperatures and large expansion velocities.Comment: 30 pages, 13 figures, Latex using documentstyle[12pt,a4,epsfig], to appear in Eur. Phys. J.