We have completed our introduction to ‘braided geometry’ with the basic constructions and basic examples. We are now ready to apply these methods back to ordinary Hopf algebras. The construction in this lecture is called ‘bosonisation’ because it turns a braided group (which is a generalisation of a supergroup) into an ordinary Hopf algebra (such as an ordinary group) [the term comes from physics].

First of all, a general concept which we will use. It is just for clarity (we are not going to do any heavy category theory). Thus, a monoidal functor F : C → V between monoidal categories is a pair (F : c) where

1. F is a functor.

2. c: F2 → F ∘ × is a natural isomorphism between the functors F2, F ∘ × : C×C → V (here F2(V, W) = F(V) × F(W)). This is a collection of functorial isomorphisms

3. The condition in Figure 16.1 holds.

4. We also require

for compatibility with the unit object.

This is more or less obvious, and in fact in our present applications, Φ will be the trivial vector space associativity and c will also be the trivial vector space identification. So we are just saying that F respects ×.

Lemma 16.1

Let H be a quasitriangular Hopf algebra. There is a monoidal functor

for all v ∈ V. Here (β is said to be the coaction ‘induced’ by an action on any module V.

Proof This is a nice exercise from the axioms of a quasitriangular structure. That (β is indeed a coaction is the (id × δ)R axiom.