Rigid indecomposable modules and real Schur roots

Ponente: Christof Geiss
Institución: IM-UNAM
Tipo de Evento: Researcher, Human Resource Training

When	Mar 22, 2019 from 10:30 AM to 11:30 AM
Where	Auditorio Nápoles Gándara
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(Joint with B. Leclerc and J. Schröer).

Let $C$ be a symmetrizable generalized Cartan matrix with symmetrizer $D$ and orientation $\Omega$. In previous work we constructed for any field $\FF$ an $\FF$-algebra $H := H_\mathbb{FF}(C,D,\Omega)$, defined in terms of a quiver with relations, such that the locally free $H$-modules behave in many aspects like representations of a hereditary algebra $\tilde{H}$ of the corresponding type.

We introduce a Noetherian algebra $\hat{H}$ over a power series ring, which provides a direct link between the representation theory of $H$ andof $\tilde{H}$. There are reduction and a localization functors relating the module categories of $\hat{H}$, $\tilde{H}$ and $H$.

These are used to show that there are natural bijections between the sets of isoclasses of tilting modules over the three algebras $\hat{H}$, $\ilde{H}$ and $H$. This allows us to show that the indecomposable rigid locally free $H$-modules are parametrized, via their rank vector, by the real Schur roots associated to $(C,\Ome)$. Moreover, the left finite bricks of $H$, in the sense of Asai, are parametrized, via their dimension vector, by the real Schur roots associated to $(C^T,\Omega)$.