We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzezinski and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. We study the bosonisation of any braided group provides as a trivial principal bundle in three ways.

We construct double biproduct, bicrossproduct, double crossproduct, double bicrossproduct Lie bialgebras from braided Lie bialgebras. The relations between them are found. The main result generalizes Majid’s matched pair of Lie algebras, Drinfeld’s quantum double of Lie bialgebras, and Masuoka’s cross product Lie bialgebras. Some properties of double biproduct Lie bialgebras are given. In the a...

We introduce a notion of " freely braided element " for simply laced Cox-eter groups. We show that an arbitrary group element w has at most 2 N (w) com-mutation classes of reduced expressions, where N(w) is a certain statistic defined in terms of the positive roots made negative by w. This bound is achieved if w is freely braided. In the type A setting, we show that the bound is achieved only f...

Composites reinforced with woven or braided textiles exhibit high structural stability and excellent damage tolerance thanks to yarn interlacing. With their high stiffness-to-weight and strength-to-weight ratios, braided composites are attractive for aerospace and automotive components as well as sports protective equipment. In these potential applications, components are typically subjected to...

Braided Hopf algebras have attracted much attention in both mathematics and mathematical physics (see e.g. [1][4][13][15][17][16][20][23]). The classification of finite dimensional Hopf algebras is interesting and important for their applications (see [2] [22]). Braided Hopf algebras play an important role in the classification of finite-dimensional pointed Hopf algebras (e.g. [2][1] [19]). The...

We show that the differential complex Ω B over the braided matrix algebra BM q (N) represents a covariant comodule with respect to the coaction of the Hopf algebra Ω A which is a differential extension of GL q (N). On the other hand, the algebra Ω A is a covariant braided comodule with respect to the coaction of the braided Hopf algebra Ω B. Geometrical aspects of these results are discussed.

Recently, Green and Losonczy [5, 6] introduced freely braided permutations as a special class of restricted permutations that has arisen in the study of Schubert varieties. They suggest as an open problem to enumerate the number of freely braided permutations in Sn. In this paper, we prove that the generating function for the number of freely braided permutations in Sn is given by 1−3x−2x2 +(1+...

The channels of multi-threaded (braided) rivers occur commonly in areas that fulfill certain conditions such as substantial influx bed material and gradients large enough to create significant energy flowing water. Natural favoring the formation multithreaded are present Carpathians Poland Podhale – a basin located piedmont area high-mountain Tatra massif. had experienced glaciation Pleistocene...

The Benin Formation is well exposed along the Atamiri River, which traverses Uli town, in the Niger Delta Basin of Southeastern Nigeria. Lithologic sections were measured, described and sampled so as to decipher the paleodepositional environment, using an integration of lithofacies data, pebble morphology, grain size and paleocurrent analysis. The lithofacies study suggests that the Benin Forma...