## The Diagonal of the Stasheff polytope

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@MISC{Loday_thediagonal,

author = {Jean-louis Loday},

title = {The Diagonal of the Stasheff polytope},

year = {}

}

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### Abstract

We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with analogous constructions due to Saneblidze-Umble and Markl-Shnider based on the Boardman-Vogt cubical decomposition of the Stasheff polytope.

### Citations

322 |
Homotopy associativity of H-spaces
- Stasheff
- 1963
(Show Context)
Citation Context ... AA-infinity algebra, diagonal. Introduction An associative algebra up to homotopy, or A∞-algebra, is a chain complex (A, dA) equipped with an n-ary operation µn for each n ≥ 2 verifying µ◦µ = 0. See =-=[15]-=-, or, for instance, [5]. Here we put µ := dA + µ2 + µ3 + · · · : T (A) → T (A), where µn has been extended to the tensor coalgebra T (A) by coderivation. In particular µ2 is not associative, but only ... |

191 |
Homotopy invariant algebraic structures on topological spaces
- Boardman, Vogt
- 1973
(Show Context)
Citation Context ...n−1 simp into a new simplicial set fatKn−1 simp , cf. [8]. Then Kn simp is defined as the cone over fatK n−1 simp (as in the original construction of Stasheff [15]). For n = 1, we have K1 simp = K1 = =-=[0, 1]-=- (the interval). Examples: K2 simp and fatK3 simp �� �� �� �� �� � �������� � ��������� ��� � b �� �� � �� �� � �� � �� �� � � a � � � � � � � ��������������� ���� c ���� � �� �� �� �� �� �� � � � ���... |

131 | Discriminants, resultants & multidimensional determinants. Birkhauser - Gelfand, Kapranov, et al. - 2008 |

116 |
Operads in algebra, topology and physics
- Markl, Shnider, et al.
- 2002
(Show Context)
Citation Context ...hism from the operad A∞ governing the associative algebras up to homotopy to the operad AA∞ is deduced from the simplicial structure of the associahedron. 2.1 Differential graded non-symmetric operad =-=[10]-=- By definition a differential graded non-symmetric operad, dgns operad for short, is a family of chain complexes Pn = (Pn, d) equipped with chain complex morphisms8 Jean-Louis Loday γi1···in : Pn ⊗ P... |

87 | Tensor Construction of Open String Theories I: Foundations, Nucl. Phys. B505
- Gaberdiel, Zwiebach
- 1997
(Show Context)
Citation Context ...in the following sense: µ2 ◦ (µ2 ⊗ id) − µ2 ◦ (id ⊗ µ2) = dA ◦ µ3 + µ3 ◦ d A ⊗3 . Putting an A∞-algebra structure on the tensor product of two A∞-algebras is a long standing problem, cf. for instance =-=[12, 2]-=-. Recently a solution has2 Jean-Louis Loday been constructed by Saneblidze and Umble, cf. [13, 14], by providing a diagonal A∞ → A∞ ⊗ A∞ on the operad A∞ which governs the A∞-algebras. Recall that, o... |

79 |
Grundlehren der Mathematischen Wissenschaften 114
- Maclane, Homology
- 1963
(Show Context)
Citation Context ...simp complex is the normalized chain complex of the simplicial set. It is the quotient of the chain complex associated to the simplicial set, divided out by the degenerate simplices (cf. for instance =-=[9]-=- Chapter VIII). A basis of C0(Kn simp ) is given by P BTn+2 and a basis of Cn(Kn simp ) is given by the (n + 1)n−1 top simplices (in bijection with the parking functions, cf. [8]). It is zero higher u... |

69 | The associahedron and triangulations of the n-gon - Lee - 1989 |

66 | Introduction to A-infinity algebras and modules
- Keller
(Show Context)
Citation Context ...agonal. Introduction An associative algebra up to homotopy, or A∞-algebra, is a chain complex (A, dA) equipped with an n-ary operation µn for each n ≥ 2 verifying µ◦µ = 0. See [15], or, for instance, =-=[5]-=-. Here we put µ := dA + µ2 + µ3 + · · · : T (A) → T (A), where µn has been extended to the tensor coalgebra T (A) by coderivation. In particular µ2 is not associative, but only associative up to homot... |

53 | Dialgebras and Related Operads - Loday, Dialgebras - 2001 |

45 | Polytopal realizations of generalized associahedra
- Chapoton, Fomin, et al.
- 2002
(Show Context)
Citation Context ...spaces. It is a convex polytope of dimension n with one vertex for each planar binary tree with n + 2 leaves. There are various realizations of Kn as a polytope in the literature either published cf. =-=[1]-=-, [2], [3], [8], [9], or unpublished (D. Grayson, M. Haiman). Here we propose a simple one, which has the following advantages, on top of being simple: • it respects the symmetry, • it fits with the c... |

36 | Relating the associahedron and the permutohedron, Operads
- Tonks
- 1997
(Show Context)
Citation Context ...ex polytope of dimension n with one vertex for each planar binary tree with n + 2 leaves. There are various realizations of Kn as a polytope in the literature either published cf. [1], [2], [3], [8], =-=[9]-=-, or unpublished (D. Grayson, M. Haiman). Here we propose a simple one, which has the following advantages, on top of being simple: • it respects the symmetry, • it fits with the classical realization... |

32 | Umble R., Diagonals on the permutahedra, multiplihedra and associahedra
- Saneblidze
(Show Context)
Citation Context ... (21) + (22) ) +(03) ⊗ ( (11)(21) + (12)(21) + (11)(22) ) + µ5 ⊗ (11)(21)(31) 03 b 0 + 0 + 0 + 0 03 c 0 + 0 + 0 + 0 02 α 0 + 0 + (02) ⊗ ( (11)(31) + (12)(31) ) + 0 02 β 0 + (01)(02) ⊗ ( (11) + (12) + =-=(13)-=- ) + 0 + 0 01 a 0 + (01)(01) ⊗ (31) + (01) ⊗ (21)(31) 01 b 0 + 0 + 0 + 0 01 c 0 + 0 + 0 + 0 12 α 0 + 0 + (12) ⊗ ( (12)(21) + (11)(22) ) + 0 12 β 0 + 0 + 0 + 0 11 a 0 + 0 + (11) ⊗ (12)(31) + 0 11 b 0 +... |

27 | Order structure on the algebra of permutations and of planar binary trees
- Loday, Ronco
(Show Context)
Citation Context ...rve that the sum of the n coordinates of M(σ) is S(n), hence all the points M(σ) lie in the affine hyperplane H. Under interpreting a permutation as a planar binary tree with levels (cf. for instance =-=[LR]-=-), and then forgetting the levels, one gets a welldefined map ψ : Sn→ Yn. ❅�� ❅� ❅ For instance ψ(1 2 3) = �� �� � � ❅�� ❅� � }. ❅�� ❅ ❅ , ψ(1 3 2) = ψ(2 3 1) = 1.2 Proposition. The Stasheff polytope ... |

20 |
Tessellations of moduli spaces and the mosaic operad, in Homotopy invariant algebraic structures
- Devadoss
- 1998
(Show Context)
Citation Context ...s. It is a convex polytope of dimension n with one vertex for each planar binary tree with n + 2 leaves. There are various realizations of Kn as a polytope in the literature either published cf. [1], =-=[2]-=-, [3], [8], [9], or unpublished (D. Grayson, M. Haiman). Here we propose a simple one, which has the following advantages, on top of being simple: • it respects the symmetry, • it fits with the classi... |

12 |
From operads to “physically” inspired theories, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy
- Stasheff
- 1995
(Show Context)
Citation Context ...n can be obtained by truncating the standard simplex along some hyperplanes, one per each cell of ∆ n−1 (except the big cell). Truncating only along “admissible” ones gives the Stasheff polytope (cf. =-=[St2]-=-). We give the explicit equations of these hyperplanes. As a consequence we get the results announced in the first section. 2.1 Shuffles and hyperplanes. The intersection of the hyperplane H with the ... |

8 |
cellular W-construction and products of A∞-algebras preprint arXiv:math/0312277v1
- Markl, Shnider, et al.
(Show Context)
Citation Context ... the operad A∞ is the minimal model of the operad As governing the associative algebras. The differential graded module (A∞)n of the n-ary operations is the chain complex of the Stasheff polytope. In =-=[10]-=- Markl and Shnider give a conceptual construction of the Saneblidze-Umble diagonal by using the Boardman-Vogt model of As. This model is the bar-cobar construction on As, denoted ΩBAs, in the operadic... |

6 | Homotopy associativity of H spaces - Stasheff - 1963 |

4 | Parking functions and triangulation of the associahedron
- Loday
- 2007
(Show Context)
Citation Context ...→ ΩBAs → ΩBAs ⊗ ΩBAs p⊗p −→ A∞ ⊗ A∞ . The aim of this paper is to give an alternative solution to the diagonal problem by relying on the simplicial decomposition of the Stasheff polytope described in =-=[8]-=- and using the diagonal of the standard simplex. It leads to a new model AA∞ of the operad As, whose dg module (AA∞)n is the chain complex of a simplicial decomposition of the Stasheff polytope. Becau... |

4 |
Die Grundlehren der mathematischen wissenschaften, Band 114
- Maclane
(Show Context)
Citation Context ...simp complex is the normalized chain complex of the simplicial set. It is the quotient of the chain complex associated to the simplicial set, divided out by the degenerate simplices (cf. for instance =-=[9]-=- Chapter VIII). A basis of C0(Kn simp ) is given by PBTn+2 and a basis of Cn(Kn simp ) is given by the (n + 1)n−1 top simplices (in bijection with the parking functions, cf. [8]). It is zero higher up... |

3 |
On Tamari lattices
- Geyer
- 1994
(Show Context)
Citation Context ...e canonical orientation of the cube coincides precisely with the Tamari poset structure. The referee informed me that these pictures already appeared (without any mention of the Stasheff polytope) in =-=[3]-=-. Surprisingly, this way of viewing the associahedron is related to an algebraic structure on the set of planar binary trees P BT = ⋃ n≥1 P BTn, related to dendriform algebras. Indeed there is a non-c... |

2 | The twisted Cartesian model for the double path fibration. ArXiv Mathematics e-prints
- Kadeishvili, Saneblidze
- 2002
(Show Context)
Citation Context ...� ���� �� ���� �� �� �� �� �� ↦→ � ������� ������� �� �� �� �� The exact way the main cube is deformed is best explained by drawing the associahedron on the cube. This is recalled in the Appendix. In =-=[4]-=- Kadeishvili and Saneblidze give a general method for constructing a diagonal on some polytopes admitting a cubical decomposition along the same principle (inflating the main cube). Markl and Shnider ... |

2 |
cellular W -construction and products of A∞-algebras
- Markl, Shnider, et al.
(Show Context)
Citation Context ...hain complex of the Stasheff polytope. The method of Saneblidze and Umble consists in providing an explicit (i.e. combinatorial) diagonal of the Stasheff polytope considered as a cellular complex. In =-=[11]-=- Markl and Shnider give a construction of the Saneblidze-Umble diagonal by using the Boardman-Vogt model of As. This model is the bar-cobar construction on As, denoted ΩBAs, in the operadic framework.... |

2 |
multidimensional determinants. In: Mathematics: Theory and Applications
- Gelfand, Kapranov, et al.
- 1994
(Show Context)
Citation Context ... is a convex polytope of dimension n with one vertex for each planar binary tree with n + 2 leaves. There are various realizations of Kn as a polytope in the literature either published cf. [1], [2], =-=[3]-=-, [8], [9], or unpublished (D. Grayson, M. Haiman). Here we propose a simple one, which has the following advantages, on top of being simple: • it respects the symmetry, • it fits with the classical r... |

1 |
A∞-structures, modèle minimal de Baues-Lemaire et homologie des fibrations, Thèse d’Etat
- Prouté
- 1984
(Show Context)
Citation Context ...n we give its image (up to signs) as a sum of four terms, since in the AW morphism there are four terms.24 Jean-Louis Loday 03 a (01)(01)(01) ⊗ µ5 + (02)(01) ⊗ ( − (21) + (22) ) +(03) ⊗ ( (11)(21) + =-=(12)-=-(21) + (11)(22) ) + µ5 ⊗ (11)(21)(31) 03 b 0 + 0 + 0 + 0 03 c 0 + 0 + 0 + 0 02 α 0 + 0 + (02) ⊗ ( − (11)(31) − (12)(31) ) + 0 02 β 0 + (01)(02) ⊗ ( (11) + (12) + (13) ) + 0 + 0 01 a 0 + (01)(01) ⊗ (31... |

1 |
A Diagonal on the Associahedra, preprint, ArXiv math.AT/0011065
- Saneblidze, Umble
(Show Context)
Citation Context ...gebra structure on the tensor product of two A∞-algebras is a long standing problem, cf. for instance [12, 2]. Recently a solution has2 Jean-Louis Loday been constructed by Saneblidze and Umble, cf. =-=[13, 14]-=-, by providing a diagonal A∞ → A∞ ⊗ A∞ on the operad A∞ which governs the A∞-algebras. Recall that, over a field, the operad A∞ is the minimal model of the operad As governing the associative algebras... |