Non-kissing and non-crossing complexes for locally gentle algebras

Abstract

Starting from a locally gentle bound quiver, we define on the one hand a simplicial complex, called the non-kissing complex. On the other hand, we construct a punctured, marked, oriented surface with boundary, endowed with a pair of dual dissections. From those geometric data, we define two simplicial complexes: the accordion complex, and the slalom complex, generalizing work of A. Garver and T. McConville in the case of a disk. We show that all three simplicial complexes are isomorphic, and that they are pure and thin. In particular, there is a notion of mutation on their facets, akin to τ -tilting mutation. Along the way, we also construct inverse bijections between the set of isomorphism classes of locally gentle bound quivers and the set of homeomorphism classes of punctured, marked, oriented surfaces with boundary, endowed with a pair of dual dissections.YP, VP and PGP were partially supported by the French ANR grant SC3A (15 CE40 0004 01). VP was partially supported by the French ANR grant CAPPS (17 CE40 0018). 1 arXiv:1807.04730v2 [math.CO] 13 Jul 20182.1. Locally gentle bound quivers and their blossoming quivers. We consider a bound quiverQ = (Q, I), formed by a finite quiver Q and an ideal I of the path algebra kQ (the k-vector space generated by all paths in Q, including vertices as paths of length zero, with multiplication induced by concatenation of paths) such that I is generated by linear combinations of paths of length at least two. Note that we do not require that the quotient algebra kQ/I be finite dimensional. The following definition is adapted from [BR87].Definition 2.1. A locally gentle bound quiverQ := (Q, I) is a (finite) bound quiver where (i) each vertex a ∈ Q 0 has at most two incoming and two outgoing arrows, (ii) the ideal I is generated by paths of length exactly two, (iii) for any arrow β ∈ Q 1 , there is at most one arrow α ∈ Q 1 such that t(α) = s(β) and αβ / ∈ I (resp. αβ ∈ I) and at most one arrow γ ∈ Q 1 such that t(β) = s(γ) and βγ / ∈ I (resp. βγ ∈ I). The algebra kQ/I is called a locally gentle algebra. A gentle bound quiver is a locally gentle bound quiverQ such that the algebra kQ/I is finite-dimensional; the algebra is then called a gentle algebra.Definition 2.2. A locally gentle bound quiverQ is complete if any vertex a ∈ Q 0 is incident to either one (a is a leaf) or four arrows (a is an internal vertex). The pruned subquiver of a

τ-tilting finite gentle algebras are representation-finite

“…They were introduced in [AS87] in the study of iterated tilted algebras of type A m , but have recently appeared in connection with dimer models [Boc12,Bro12], enveloping algebras of some Lie algebras [HK06], cluster algebras and categories arising from triangulated surfaces [LF09,ABCJP10], m-Calabi-Yau tilted algebras [GE17,GE18], non-kissing complexes of grids and associated objects [McC17, GM18, PPP17, BDM + 17], non-commutative nodal curves [BD18], and partially wrapped Fukaya categories [HKK17,LP18]. Surface models have been introduced to study the category representations of a gentle algebra and associated categories [BCS18,OPS18,PPP18].…”

On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms

“…Using seminal results from [3,4], we show that, rather strikingly, certain projective n−4 2 forms on kinematic space associahedra define planar amplitudes for massless quartic interactions. The notion of projectivity is the same as the one introduced in [2] and is accomplished in this case by the dependence of these forms on certain quadrangulations Q of an n-gon.…”

“…Although the construction in [4] relies on inputs from theory of quivers and corresponding path algebras [3], the convex realisation of Stokes polytope can be understood without a detailed knowledge of them. In the following we state the key result in [4] pertaining to such realisations (theorem 2.44 in [4]) and explain the construction "algorithmically" .…”

“…• We first draw hollow vertices v 1 , v 3 , v 4 , v 5 , v 6 and v 8 on the edges (1,2), (3,4), (4,5), (5,6), (6,7) and (8,1) respectively. We place a hollow vertex v 1,4 on the diagonal D 1,4 and a hollow…”

“…In this paper we develop various aspects of this proposal. Using recent seminal results in Representation theory [3,4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction. We generalise the results of [1] and show that for any n-particle amplitude, the positive geometry associated to the projective scattering form is a convex realisation of Stokes polytope which can be naturally embedded inside one of the ABHY associahedra defined in [2,5].…”