Fans and polytopes in tilting theory I: Foundations

For a finite dimensional algebra A over a field k, the 2-term silting complexes of A gives a simplicial complex $Δ(A)$ called the g-simplicial complex. We give tilting theoretic interpretations of the h-vectors and Dehn-Sommerville equations of $Δ(A)$. Using g-vectors of 2-term silting complexes, $Δ(A)$ gives a nonsingular fan $Σ(A)$ in the real Grothendieck group K₀(proj A)R called the g-fan. We give several basic properties of $Σ(A)$ including sign-coherence, sign decomposition, idempotent reductions, Jasso reductions, pairwise positivity and a connection with Newton polytopes of A-modules. Moreover, $Σ(A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in K₀(proj A)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.