Preface The term " quantum group " was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics and physics. The development of Hopf algebras can be divided into five stages. The first...

We establish a general coherence theorem for lax operad actions on an n-category which implies that an n-category with such an action is lax equivalent to one with a strict action. This includes familiar coherence results (e.g. for symmetric monoidal categories) and many new ones. In particular, any braided monoidal n-category is lax equivalent to a strict braided monoidal n-category. We also o...

This thesis proposes a network coding based and easily-realizable composite network topologic model. It is composed of disjoint multi-path routing (DMR) and braided multi-path routing (BMR) in wireless sensor network, and network coding technology is also employed in our multi-path routing. With the use of the Node Coding technology and the multi-path technology, NC-RMR (Network Coding based Re...

We introduce an addition law for the usual quantummatrices A(R) by means of a coaddition ∆t = t⊗ 1 + 1⊗ t. It supplements the usual comultiplication ∆t = t⊗ t and together they obey a codistributivity condition. The coaddition does not form a usual Hopf algebra but a braided one. The same remarks apply for rectangular m × n quantum matrices. As an application, we construct leftinvariant vector ...

By studying the braid group action on Milnor’s construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described group.

We introduced a braided Sweedler cohomology, which is adequate to work with the H-braided cleft extensions studied in [G-G1]. Introduction In [Sw] a cohomology theory H(H,A) for a commutative module algebra A over a cocommutative Hopf algebra H was introduced. This cohomology is related to those of groups an Lie algebras in the following sense. When H is a group algebra k[G], then H(H,A) is can...

Purpose. To evaluate the efficacy and safety of polybutylate-coated braided polyester (Ethibond* 5-0) suture for levator aponeurosis suturing to the anterior tarsal surface in involutional blepharoptosis repair surgery. Methods. Ten consecutive patients (16 eyes) with acquired blepharoptosis which resulted from levator aponeurosis dehiscence with good levator function had gone through surgery a...

In this paper we construct “structural” pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand, we propose a general homology theory for pre-braided vector spaces and braided modules, based on the quantum co-shuffle comultiplication. Applied to the stru...

We consider a class of graphs embedded in R2 as noncommutative proof-nets with an explicit exchange rule. We give two characterization of such proof-nets, one representing proof-nets as CW-complexes in a two-dimensional disc, the other extending a characterization by Asperti. As a corollary, we obtain that the test of correctness in the case of planar graphs is linear in the size of the data. B...