Clusters and the preprojective algebra

I was in the UK last week at a conference and a wedding, which were both very enjoyable but understandably led to no blogging. I hope to post some of my notes from the conference at some point; Gavril Farkas gave a lovely talk on Abelian varieties in particular.

Being thrust back into the semester on my return, it seems that I have a class paper to write for a Lie groups course. I’ve wanted to explore the appearance of cluster phenomena in Lie theory for a while and this seems like a good opportunity to do so. Given a simply connected simple complex Lie group $G$ of Cartan-Killing type ADE with Lie algebra $\mathfrak{g}$ and a maximal unipotent subgroup $N$ with Lie algebra $\mathfrak{n}$, Lusztig constructed a ‘semicanonical basis’ of the enveloping algebra $U\mathfrak{n}$. This is ‘semicanonical’ because, using the duality between $U\mathfrak{n}$ and the coordinate ring $\mathbb{C}[N]$, one obtains a dual basis of $\mathbb{C}[N]$ with the property that it contains a basis for all the irreducible representations of $\mathfrak{g}$, which are well-known to live inside $\mathbb{C}[N]$.

These basis elements are generally hard to compute – involving Euler characteristics of some flag varieties – but many of them can be found using cluster mutation. $\mathbb{C}[N]$ is a cluster algebra – not usually finite-type! – and the natural cluster structure found by Fomin-Zelevinsky coincides with the dual semicanonical basis of Lusztig in the sense that the cluster variables are part of the basis. This in particular establishes that they are linearly independent, which was only conjectured in general by Fomin-Zelevinsky!

Big picture aside, one of the objects needed to make this comparison is the preprojective algebra, which turns out to be an old friend from the McKay correspondence. I find it intriguing that one of the main techniques for extracting information about hard noncommutative algebras is by viewing them as path algebras of quivers with relations – I believe that the zenith of this point of view is Auslander-Reiten theory – and this approach works excellently here. I wrote some notes setting up the preprojective algebra and discussing some of its startling features and their interplay with cluster algebras, which should be a prelude to the detail of the semicanonical basis.