Dichotomy of the Addition of Natural Numbers

Loday, Jean-Louis

doi:10.1007/978-3-0348-0405-9_4

Cited by 5 publications

(2 citation statements)

References 31 publications

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“…Remark 5.1. There are some more concrete realisations of Stasheff cells (see J. L. Loday [Lod12] for example).…”

Section: ∞ -Structure Inmentioning

confidence: 99%

Smooth $A_{\infty}$ form on a diffeological loop space

Iwase¹

2022

Preprint

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To construct an ∞ -structure for a loop space in the category of diffeological spaces, we have two major problems. Firstly, the concatenation of paths in the category of diffeological spaces needs a small technical trick (see P. I-Zemmour [IZ13]), which apparently restricts the number of iterations of concatenations. To resolve this difficulty, we use the notion of a reflexivity introduced by J. Watts [Wat12]. Secondly, we do not have a natural decomposition of an associahedron as a simplicial or a cubical complex. So we introduce a notion of a cubic complex as an extension of a simplicial or a cubical complex. Using it, we can show, for a smooth CW complex, the existence of a smooth bijection from a smooth cubic complex to the smooth CW complex. Finally, we decompose an associahedron as a cubic complex naturally, so that structure maps among associahedra are smooth. Thus the smooth loop space of a nice diffeological space is a smooth ∞ -space.

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“…Remark 5.1. There are some more concrete realisations of Stasheff cells (see J. L. Loday [Lod12] for example).…”

Section: ∞ -Structure Inmentioning

confidence: 99%

Smooth $A_{\infty}$ form on a diffeological loop space

Iwase¹

2022

Preprint

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“…It should be noticed, however, that not all faces are regular or flat quadrangles or hexagons, respectively heptagons (as also in Figure 17 with pentagons, cf. [14]).…”

Section: Some Insights Into the Case M >mentioning

confidence: 99%

KP Solitons, Higher Bruhat and Tamari Orders

Müller‐Hoissen

Dimakis

2012

Associahedra, Tamari Lattices and Related Structures

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In a tropical approximation, any tree-shaped line soliton solution, a member of the simplest class of soliton solutions of the Kadomtsev-Petviashvili (KP-II) equation, determines a chain of planar rooted binary trees, connected by right rotation. More precisely, it determines a maximal chain of a Tamari lattice. We show that an analysis of these solutions naturally involves higher Bruhat and higher Tamari orders.

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Many non-equivalent realizations of the associahedron

2014

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Abstract. Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two different ways of constructing exponential families of realizations of the n-dimensional associahedron with normal vectors in {0, ±1}n , generalizing the constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify the associahedra obtained by these constructions modulo linear equivalence of their normal fans and show, in particular, that the only realization that can be obtained with both methods is the Chapoton-FominZelevinsky (2002) associahedron.For the Hohlweg-Lange associahedra our classification is a priori coarser than the classification up to isometry of normal fans, by Bergeron-Hohlweg-Lange-Thomas (2009). However, both yield the same classes. As a consequence, we get that two Hohlweg-Lange associahedra have linearly equivalent normal fans if and only if they are isometric.The Santos construction, which produces an even larger family of associahedra, appears here in print for the first time. Apart of describing it in detail we relate it with the c-cluster complexes and the denominator fans in cluster algebras of type A.A third classical construction of the associahedron, as the secondary polytope of a convex n-gon (Gelfand-Kapranov-Zelevinsky, 1990), is shown to never produce a normal fan linearly equivalent to any of the other two constructions.

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Dichotomy of the Addition of Natural Numbers

Cited by 5 publications

References 31 publications

Smooth $A_{\infty}$ form on a diffeological loop space

Smooth $A_{\infty}$ form on a diffeological loop space

KP Solitons, Higher Bruhat and Tamari Orders

Many non-equivalent realizations of the associahedron

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