We obtain the double factorization of braided bialgebras or braided Hopf algebras, give relation among integrals and semisimplicity of braided Hopf algebra and its factors. We find an example which show that the 8th Kaplansky’s conjecture does not hold for braided Hopf algebras.

We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space L equipped with a bracket [ , ] : L⊗L → L and a Yang-Baxter operator Ψ : L⊗L → L⊗L obeying some axioms. We show that such an object has an enveloping braided-bialgebra U(L). We show that every generic R-matrix leads to such a braided Lie algebra with [ , ] given by structure constants cK determined ...

Hughes has defined a class of groups, which we call FSS (finite similarity structure) groups. Each FSS group acts on a compact ultrametric space by local similarities. The best-known example is Thompson’s group V . Guided by previous work on Thompson’s group V , we establish a number of new results about FSS groups. Our main result is that a class of FSS groups are of type F∞. This generalizes ...

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups Uq(g). They have the same FRT generators l± but a matrix braided-coproduct ∆L = L⊗L where L = l+Sl−, and are se...

We present a systematic introduction to the geometry of linear braided spaces. These are versions of R n in which the coordinates x i have braid-statistics described by an R-matrix. From this starting point we survey the author's braided-approach to q-deformation: braided differentiation, exponentials, Gaussians, integration and forms, i.e. the basic ingredients for q-deformed physics are cover...

Digital Elevation Models (DEMs) act as an essential input to hydraulic models for the simulation of flood hazard maps. Coarser resolution DEMs are subject higher levels uncertainty, particularly in capturing river-bed bathymetry. The principal aim present study was assess potential use freely available and 1D 2D HEC-RAS modeling approaches mapping four different types river morphology. DEM crea...

This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups all in analogy with superlines, superplanes etc. The main idea is that the bose-fermi ±1 statistics between Grassmannn coordinates is now replaced by a gener...

The goal of this paper is to characterise certain probability laws on a class of quantum groups or braided groups that we will call nilpotent. First we introduce a braided analogue of the Heisenberg-Weyl group, which shall serve as standard example. We introduce Gaussian functionals on quantum groups or braided groups as functionals that satisfy an analogue of the Bernstein property, i.e. that ...

We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable nondegenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theo...

We study four (a priori) different ways in which an open book decomposition of the 3-sphere can be defined to braided. These include generalised exchangeability by Morton and Rampichini mutual braiding Rudolph, were shown equivalent Rampichini, as well P-fiberedness a property related simple branched covers $S^3$ inspired work Montesinos Morton. prove that these notions braided are actually all...