Whereas the last lecture was about ‘bosonisation’ and led to the uq(b+) quantum groups, this time we will do ‘double bosonisation’ and the construction of uq(sl2) by these braided group methods. Note that this approach is a little idiosyncratic but it provides, in my opinion, the deepest insight into the construction of quantum groups of this type. This is also the last lecture with ‘braid diagrams’ – we finish up this middle section of the course. We will, however, link this approach with Lusztig's in the lecture after this one.

First of all, a few final remarks about bosonisation. Although we gave the proof quite explicitly, its real content is no more than the (quite elementary) construction of a monoidal category from B – the category BC of braided modules in C - HM. After this, it is more or less clear that there is some ordinary Hopf algebra (in our case B⋊H) whose modules are this category. The abstract formulation of this last step is called ‘Tannaka-Krein reconstruction’ – given a monoidal category C and a monoidal functor C → Vec obeying some technical conditions, there is an ordinary Hopf algebra in Vec such that the functor factors through its modules. In nice cases this identifies C with this category of modules. There is also a technically superior comodule version.

Moreover, bosonisation itself provides a kind of ‘functor’ from braided groups of a certain kind (in module or comodule categories) to ordinary Hopf algebras. There is also a partially denned ‘functor’ in the other direction.