Fans and polytopes in tilting theory I: Foundations

Abstract

For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $\Delta (A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $\Delta (A)$. Using $g$-vectors of 2-term silting complexes, $\Delta (A)$ gives a nonsingular fan $\Sigma (A)$ in the real Grothendieck group $K_0(\mathsf{proj}A{)}_{\mathbb{R}}$ called the $g$-fan. We give several basic properties of $\Sigma (A)$ including sign-coherence, sign decomposition, idempotent reductions, Jasso reductions, pairwise positivity and a connection with Newton polytopes of $A$-modules. Moreover, $\Sigma (A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in $K_0(\mathsf{proj}A{)}_{\mathbb{R}}$ called the $g$-polytope of $A$. We call $A$ $g$-convex if $P(A)$ is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of $A$. There are precisely 7 convex $g$-polyogons up to isomorphism. We give a classification of algebras whose $g$-polytopes are smooth Fano. We study $g$-fans and $g$-polytopes of two important classes of algebras. We show that the $g$-fan of a classical or generalized preprojective algebra is given by the Coxeter fan. It is $g$-convex if and only if it is of type $A$ or $B$, and in this case, its $g$-polytope is the dual polytope of the short root polytope. Moreover we classify Brauer graph algebras which are $g$-convex, and describe their $g$-polytopes as the root polytopes of type $A$ or $C$.

63 pages, v4: The old section on Jacobian algebras has been removed and will be included in a separate paper, as it requires a more general setting than other sections. v2: Fix typos. We remove the part of convex g-polygons, which will be discussed in our next paper

Cited by in corpus (1)

• Acyclic cluster algebras with dense $g$-vector fans