Fans and polytopes in tilting theory

Abstract

For a finite dimensional algebra A over a field k, the 2-term silting complexes of A gives a simplicial complex ∆(A) called the g-simplicial complex. We give tilting theoretic interpretations of the h-vectors and Dehn-Sommerville equations of ∆(A). Using g-vectors of 2-term silting complexes, ∆(A) gives a nonsingular fan Σ(A) in the real Grothendieck group K 0 (proj A) R called the g-fan. For example, the fan of g-vectors of a cluster algebra is given by the g-fan of a Jacobian algebra of a non-degenerate quiver with potential. We give several properties of Σ(A) including idempotent reductions, sign-coherence, Jasso reductions and a connection with Newton polytopes of A-modules. Moreover, Σ(A) gives a (possibly infinite and non-convex) polytope P(A) in K 0 (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.We give an explicit classification of g-convex algebras of rank 2. We classify algebras whose g-polytopes are smooth Fano. We classify classical and generalized preprojective algebras which are g-convex, and also describe their g-polytope as the dual polytopes of short root polytopes of type A and B. We also classify Brauer graph algebras which are g-convex, and describe their g-polytopes as root polytopes of type A and C. Contents 1. Introduction 1 2. Preliminaries 6 3. g-simplicial complexes 9 4. g-fans 14 5. g-polytopes, c-polytopes and Newton polytopes 22 6. Convex g-polygons 30 7. Smooth Fano g-polytopes 37 8. Preprojective algebras and Coxeter fans 40 9. Jacobian algebras and Cluster algebras 49 10. Brauer graph algebras and Root polytopes 54 Acknowledgments 67 References 67

On τ-tilting finite simply connected algebras

“…Then, a similar proof for arbitrary algebras is given in [AoW,Proposition 2.20]. See also [AHIKM,Theorem 4.28] for a proof in terms of g-fans.…”