Sunday, January 13, 2013

Bosonisation of Quantum Groups, an Introduction

Let $H$ be a quasitriangular (or braided) Hopf algebra and let $\mathcal{C} = _H\mathcal{M}$ be it's category of modules (that is, a category where the objects are $H$-modules and the morphisms are $H$-linear maps between them.) Let $B \in \mathcal{C}^0$ be a Hopf algebra such that it's structure maps $m_B : B\otimes B \rightarrow B$, $\Delta_B : B \rightarrow B \otimes B$, et cetera, and all their respective objects, are all morphisms and objects in the category.

Naturally, $B$ will have it's own modules. But what if we restrict the $B$-modules we want to look at? In fact, let's only look at $B$-modules in $\mathcal{C}$, we'll call these the braided $B$-modules (the reason for this language will become clear later on). More precisely, let $_B\mathcal{C}$ be the sub-category of $B$-modules in $\mathcal{C}$ such that the action of $B$ on the module $V$, $B \otimes V \rightarrow V$, is a morphism in the category.

Does there exist a Hopf algebra $\mathcal{H}$ such that it's category of modules is exactly $_B\mathcal{C}$ ? The answer is yes. From a general abstract nonsense point of view, the affirmation comes from the Tannakian reconstruction theorems which allows us to construct certain algebraic objects from its category of modules. This is a sort of duality between the object and it's category of modules. I don't know enough about this sort of thing to comment any more.

But for us, we'll answer this question by means of Majid's theory of bosonisation of quantum groups. We won't use higher category theory, rather, we will construct the Hopf algebra $\mathcal{H}$ explicitly from $H$ and $B$, and call this construction the bosonisation of $B$. To do this, I need to study a few things:

  1. Braided Categories. When $H$ is quasitriangular, it's module category has an additional structure, a braiding, that will color all calculations in $_H\mathcal{M}$ and $_B\mathcal{C}$. I barely know what a braided category is at the moment, so I'll have to cover this in a blog post or two.
  2. Hopf algebras in braided categories. (aka ``braided groups'' in Majid's lexicon) A more concise way of describing my choice of $B$ is to say that $B$ is a braided group in $_M\mathcal{H}$. All the structure maps of $B$ will have to contend with the braiding from $H$, hence why we called $_B\mathcal{C}$ the category of braided modules. I'll study these objects in a subsequent blog post, too.
  3. Constructing the bosonisation $\mathcal{H} = B \rtimes H$ as a semidirect product of $B$ and $H$, with its structure maps built from the various morphisms in $\mathcal{C}$

The next step is to repeat this construction for a dual quasitriangular Hopf algebra to arrive at a theory of co-bosonisation, and then to repeat the process for double bosonisation.

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