This week I attended AustMS, but I still had some time on the long tram trips to do a bit of study. I mostly spent this time learning what a Hall algebra is and studying the example of nilpotent representations for the Jordan quiver. Thus far, my best summary would be that the Hall algebra of a suitable abelian category is an associative algebra which contains information about the short exact sequence structure of the category.

Let $\mathcal{A}$ be an abelian category. To make the objects of $\mathcal{A}$ into an associative $k$-algebra tracking short exact sequences, one natural idea is to take the free $k$-vector space $\mathbb{H}_{\mathcal{A}} = \bigoplus_{M \in \textnormal{Ob}(\mathcal{A})}kM$ with basis the objects of $\mathcal{A}$, and to define the product on basis elements by

$M*N = \sum\limits_{\text{extensions } K \text{ of } M \text{ by } N} a^K_{MN}K \ \text{ for constants } a^K_{MN} \in k.$

We use the direct sum to ensure that calculating products and sums requires determining finitely many entries. Our goals for this definition are that:

• The choices of structure constants $a^K_{MN}$ make the multiplication associative, and
• The structure constants record properties of the short exact sequences/extensions in $\mathcal{A}$.

With this in mind, letting
$a^K_{MN} = \#\{ \phi \in \text{Ext}(M,N) \ | \ \phi \text{ has middle term K } \}$
allows the coefficients to enumerate extensions.