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...

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...

We study the braided monoidal structure that the fusion product induces on the abelian category Wp-mod, the category of representations of the triplet W -algebra Wp. The Wp-algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalise the methods for computing fusion products, developed by Nahm,...

We classify various types of graded extensions a finite braided tensor category $$\mathcal {B}$$ in terms its 2-categorical Picard groups. In particular, we prove that by group A correspond to monoidal 2-functors from the (consisting invertible central -module categories). Such functors can be expressed Eilnberg-Mac Lane cohomology. describe detail groups symmetric fusion categories and pointed...

BACKGROUND The use of ethylene-vinyl alcohol copolymer for liquid embolization of cranial vascular lesions has resulted in microcatheter fragments entrapped in patients following endovascular procedures. Undergoing subsequent diagnostic MRI examinations poses a safety concern due to the possibility of radiofrequency heating of the metallic braid incorporated into the microcatheter. Heating of n...