We study a fan consisting of all g-vector cones associated with two-term presilting complexes of a complete gentle algebra. We show that any complete gentle algebra is g-tame, by definition, the closure of a geometric realization of its fan is the entire ambient vector space. Our main ingredients are their geometric descriptions and their asymptotic behavior under Dehn twists. As a consequence, we also get the g-tameness of a class of special biserial algebras containing Brauer graph algebras.

To study the set of torsion classes of a finite dimensional basic algebra, we use a decomposition, called sign-decomposition, parametrized by elements of {±1} n where n is the number of simple modules. If A is an algebra with radical square zero, then for each ǫ ∈ {±1} n there is a hereditary algebra A ! ǫ with radical square zero and a bijection between the set of torsion classes of A associated to ǫ and the set of faithful torsion classes of A ! ǫ . Furthermore, this bijection preserves the property of being functorially finite. As an application in τ -tilting theory, we prove that the number of support τ -tilting modules over Brauer line algebras (resp. Brauer odd-cycle algebras) having n edges is 2n n (resp. 2 2n−1 ).

We compute the number of two-term tilting complexes over an arbitrary symmetric algebra with radical cube zero.

No 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