On defining ideals or subrings of Hall algebras (with an appendix by Andrew Hubery)

Abstract

Let $A$ be a finitary algebra over a finite field $k$, and $A$-$mod$ the category of finite dimensional left $A$-modules. Let $\mathcal{H}(A)$ be the corresponding Hall algebra, and for a positive integer $r$ let $D_{r}(A)$ be the subspace of $\mathcal{H}(A)$ which has a basis consisting of isomorphism classes of modules in $A$-$mod$ with at least $r+1$ indecomposable direct summands. If $A$ is hereditary of type $A_{n}$, then $D_{r}(A)$ is known to be the kernel of the map from the twisted Hall algebra to the quantized Schur algebra indexed by $n+1$ and $r$. For any $A$, we determine necessary and sufficient conditions for $D_{r}(A)$ to be an ideal and some conditions for $D_{r}(A)$ to be a subring of $\mathcal{H}(A)$. For $A$ the path algebra of a quiver, we also determine necessary and sufficient conditions for $D_{r}(A)$ to be a subring of $\mathcal{H}(A)$.