Real Schur roots and rigid indecomposable representations

Mardi, 4. décembre 2018 - 14:00 - 15:00

Christof Geiß (Mexique)


This a report on joint work with B. Leclerc and J. Schröer. Let $A$ be a finite dimensional hereditary $K$-algebra. From its Grothendieck group $K_0(A)$, equipped with the usual homological bilinear form, one can extract  basic combinatorial invariants $(C,D,\Omega)$, where $C$ is a symmetrizable generalized Cartan matrix, $D$ a (left) symmetrizer of $C$ and $\Omega$ and orientation for $C$. Note  that $C$ will be symmetric if $K$ is algebraically closed. From Kac's theorem and its relatives it follows that for many finite dimensional hereditary algebras the classes of the finite dimensional indecomposable $A$-modules correspond precisely to the positive roots of the corresponding Kac-Moody Lie algebra $\mathfrak{g}(C)$. This holds in particular if $K$ is algebraically closed or finite, or if $C$ is not indefinite.

The real Schur roots are a subset of the positive real roots of $\mathfrak{g}(C)$, which can be defined combinatorially in terms of non-crossing partitions for $(C,\Omega)$. By work of Ringel and Crawley-Boevey they are precisely the classes of the rigid indecomposable $A$-modules for any finite dimensional hereditary $K$-algebra $A$ with combinatorial invariant $(C,D,\Omega)$.

In previous work we introduced for any field $\mathbb{F}$ an 1-Iwanaga Gorenstein $\mathbb{F}$-algebra $H:=H_\mathbb{F}(C,D,\Omega)$, defined in terms of a quiver with relations such that the so-called locally free $H$-modules behave in many aspects like the modules over an hereditary algebra with the same combinatorial invariants. We can show now that the rank vectors of the indecomposable rigid locally free $H$-modules correspond precisely to the real Schur roots associated to $(C,\Omega)$, and the dimension vectors of the left finite $H$-modules with trivial endomorphism ring correspond to the real Schur roots associated to $(C^T,\Omega)$. For the proof of this result we construct a $\mathbb{F}[\![\epsilon]\!]$-order, which allows us to relate $H$ with a particular hereditary $\mathbb{F}(\!(\epsilon)\!)$-algebra.