We give an overview of a new kind symmetry in physics which exists between observables and states and which is made possible by the language of Hopf algebras and quantum geometry. It has been proposed by the author as a feature of Planck scale physics. More recent work includes corresponding results at the semi-classical level of Poisson-Lie groups and at the level of braided groups and braided...

Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided version of Radford’s formula for the fourth power of the antipode is obtained. Connections of integ...

The machinery of braided geometry introduced previously is used now to construct the ǫ ‘totally antisymmetric tensor’ on a general braided vector space determined by R-matrices. This includes natural q-Euclidean and q-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as ̃ SOq(4) or ̃ SOq(1, 3). The Hodge ∗ operator and differentials are also cons...

The machinery of braided geometry introduced previously is used now to construct thètotally antisymmetric tensor' on a general braided vector space determined by R-matrices. This includes natural q-Euclidean and q-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as g SO q (4) or g SO q (1; 3). The Hodge operator and diierentials are also constru...

The aim of this study was to characterize the mechanical and self-expansion properties of braided bioabsorbable stents. In total four different stents were manufactured from PLLA fibres using a braiding technique. The changes in radial pressure stiffness and diameter recovery of the stents were determined initially, and after insertion and release from a delivery device. The braided stents were...

In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel′d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the “braided” aspects of YD module structure, exhibi...

Abstract The Braiding Index ( BI ), defined as the average count of intercepted channels per cross‐section, is a widely used metric for characterizing multi‐thread river systems. However, it does not account diversity (e.g., in terms water discharge) within different cross‐sections, omitting important information related to system complexity. Here we present modification BI, Entropic eBI which ...

Various attempts to find a proper generalization of the notion of Lie algebra associated to a vector space V endowed with a non symmetric braiding c appeared in the literature. In this direction, we introduce and investigate a notion of braided Lie algebra (and the associated universal enveloping algebra) which turns out to be effective for the class of braided vector spaces (V, c) whose Nichol...

We discuss and compare the notions of braided and coboundary monoidal categories. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the cactus group. We focus on the categories of representations of quantum groups and crystals and explain how while the former is a braided monoidal category, this structure does not pas...