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

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

2006-08-28

arxiv.org/abs/math/0608677v2

Mathematics

Representation Theory

9pages, also available at http://learn.tsinghua.edu.cn:8080/2002315664/dyang.htm

Scientific paper

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 $\mathcal{H}(A)$.

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