This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, as well as colour-Lie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied, the notion of ‘braided-commutative’ or ‘braided-cocommutative’ Hop...

Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras. Abstract We show that if g Γ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first order differential calculus over a co-quasitriangular Hopf algebra (A, r), then a certain extension of it is a braided Lie algebra in the category of A-comodules. This...

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonisation, associated to every braided group B in the category of Hmodules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B∗ as the ‘negative root space’ of U(B). The choice B = f recovers Lusztig’s construction of U...

We obtain new family of quasitriangular Hopf algebras C 0|n q >⊳·Uq(sun)· ⊲<C q via the author’s recent double-bosonisation construction for new quantum groups. They are versions of Uq(sun+1) with a fermionic rather than bosonic quantum plane of roots adjoined to Uq(sun). We give the n = 2 case in detail. We also consider the anyonic-double of an anyonic (Z/nZ-graded) braided group and the doub...

We develop a general theory of ‘quantum’ diffeomorphism groups based on the universal comeasuring quantum group M(A) associated to an algebra A and its various quotients. Explicit formulae are introduced for this construction, as well as dual quasitriangular and braided R-matrix versions. Among the examples, we construct the q-diffeomorphisms of the quantum plane yx = qxy, and recover the quant...

The paper begins by giving an algebraic structure on a set of coset representatives for the left action of a subgroup on a group. From this we construct a non-trivially associated tensor category. Also a double construction is given, and this allows the construction of a non-trivially associated braided tensor category. In this category we explicitly reconstruct a braided Hopf algebra, whose re...

The faithful quasi-dual H and strict quasi-dual H ′ of an infinite braided Hopf algebra H are introduced and it is proved that every strict quasi-dual H ′ is an HHopf module. The connection between the integrals and the maximal rational Hsubmodule H of H is found. That is, H ∼= ∫ l H ⊗H is proved. The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld categor...

We describe pure braided versions of Thompson’s group F . These groups, BF and B̂F , are subgroups of the braided versions of Thompson’s group V , introduced by Brin and Dehornoy. Unlike V , elements of F are order-preserving self-maps of the interval and we use pure braids together with elements of F thus preserving order. We define these groups and give normal forms for elements and describe i...

In the first part we recall two famous sources of solutions to the Yang-Baxter equation – R-matrices and Yetter-Drinfeld (=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, exh...