## A resolution (minimal model) of the PROP for bialgebras, preprint math.AT/0209007

Citations: | 17 - 3 self |

### BibTeX

@MISC{Markl_aresolution,

author = {Martin Markl},

title = {A resolution (minimal model) of the PROP for bialgebras, preprint math.AT/0209007},

year = {}

}

### OpenURL

### Abstract

Abstract. This paper is concerned with a minimal resolution of the prop for bialgebras (Hopf algebras without unit, counit and antipode). We prove a theorem about the form of this resolution (Theorem 12) and give, in Section 5, a lot of explicit formulas for the differential. Our minimal model contains all information about the deformation theory of bialgebras and related cohomology. Algebras over this minimal model are strongly homotopy bialgebras, that is, homotopy invariant versions of bialgebras.

### Citations

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Natural associativity and commutativity
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Citation Context ...n by the tensor product of linear maps, and vertical composition by the ordinary composition of maps. One can therefore imagine elements of A(m, n) as ‘abstract’ maps with n inputs and m outputs. See =-=[8, 10]-=- for precise definitions. We say that X has biarity (m, n) if X ∈ A(m, n). We will sometimes use the operadic notation: for X ∈ A(m, k), Y ∈ A(1, l) and 1 ≤ i ≤ k, we write (5) X ◦i Y := X ◦ (1 ⊗(i−1)... |

55 |
Obstructions to homotopy equivalences
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Citation Context ...refore (3) can be interpreted as a perturbation of (4) which may be informally expressed by saying that bialgebras are perturbations of 1 bialgebras. Experience 2 with homological perturbation theory =-=[3]-=- leads us to formulate: Principle. The prop B for bialgebras is a perturbation of the prop 1 1 B for 2 2bialgebras. Therefore the minimal model of the prop B for bialgebras must be a perturbation of t... |

44 |
Homotopy algebras are homotopy algebras
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Citation Context ...ver the minimal model (M, ∂) have all rights to be called strongly homotopy bialgebras, that is, homotopy invariant versions of bialgebras, as follows from principles explained in the introduction of =-=[9]-=-. This would mean, among other things, that, given a structure of a dg-bialgebra on a chain complex C∗, then any chain complex D∗, chain homotopy equivalent to C∗, has, in a certain sense, a natural a... |

38 |
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(Show Context)
Citation Context ...then where U is as in Theorem 8. Therefore the sub-1 2prop generated by ξ1 2 , ξ1 3 , ξ1 4 , . . . is in fact isomorphic to the minimal model A∞ for the operad of associative algebras as described in =-=[8]-=-. It is well-known that A∞ is the operad of cellular chains of a cellular topological operad K = {Kn}n≥1 such that each Kn is an (n − 2)-dimensional convex polyhedron – the Stasheff associahedron (see... |

36 |
Bialgebra cohomology, deformations, and quantum
- Gerstenhaber, Schack
- 1990
(Show Context)
Citation Context ...eneral philosophy, it should contain all information about the deformation theory of bialgebras. In particular, the Gerstenhaber-Schack cohomology which is known to control deformations of bialgebras =-=[2]-=- can be read off from this model as follows. Let EndV denote the endomorphism prop of V and let a bialgebra structure B = (V, µ, ∆) on V be given by a homomorphism of props β : B → EndV . The composit... |

28 | Koszul duality for dioperads
- Gan
(Show Context)
Citation Context ...or any m, n ≥ 1, a basis of the k-linear space B(m, n) as follows. Let ∈ B(1, 2) be the equivalence class, in B = Γ( , )/IB, of the generator ∈ Γ( , )(1, 2) (observe that we use the same symbol for a =-=[1]-=- generator and its equivalence class). Define := 1 ∈ B(1, 1) and, for a ≥ 2, let Let [a] := ( ⊗ 1)( ⊗ 1 ⊗2 ) · · ·( ⊗ 1 ⊗a−2 ) ∈ B(1, a). [b] ∈ B(b, 1) has the obvious similar meaning. According to [5... |

22 |
Cotangent cohomology of a category and deformations
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Citation Context ...n σ(2, 2) will be explained in Section 2). We suppose that V , as well as all other algebraic objects in this paper, are defined over a field k of characteristic zero. Let B be the k-linear prop (see =-=[6, 7]-=- for the terminology) describing bialgebras. The goal of this paper is to describe the minimal model of B, that is, a differential graded (dg) k-linear prop (M, ∂) together with a homology isomorphism... |

22 | A koszul duality for props
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- 2003
(Show Context)
Citation Context ...s by Shoikhet [20, 21, 22], and also in a recent draft by Saneblidze and Umble [19]. A general theory of resolutions of props is, besides [15], also the subject of Vallette’s thesis and its follow-up =-=[24, 25]-=-. Let us briefly sketch the strategy of the construction of our model. Consider objects (V, µ, ∆), where µ : V ⊗ V → V is an associative multiplication as in (1), ∆ : V → V ⊗ V is a coassociative comu... |

16 |
PROPped up graph cohomology
- Markl, Voronov
- 2003
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Citation Context ...draft of this paper was completed. My particular thanks are due to M. Kontsevich whose e-mail [4] shed a new light on the present work and stimulated a cooperation with A.A. Voronov which resulted in =-=[12]-=-.4 M. MARKL 2. Structure of props and 1 2 props Let us recall that a k-linear prop A (called a theory in [6, 7]) is a sequence of left-Σm right-Σn k-vector spaces {A(m, n)}m,n≥1 with compositions ◦ :... |

14 | Théorie homotopique des formes différentielles (d’aprés D - Lehmann - 1990 |

13 | On the invertibility of quantization functors
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Citation Context ...• �❅ . We will use the similar notation for elements of free props throughout the paper. All our ‘flow diagrams’ should be read from the bottom to the top. Remark 2. Enriquez and Etingof described in =-=[1]-=- a basis of the k-linear space B(m, n) for arbitrary m, n ≥ 1 as follows. Let ∈ B(1, 2) be the equivalence class, in B = Γ( , )/IB, of the generator ∈ Γ( , )(1, 2) (we use the same symbol both for a g... |

11 | The CROCs, non-commutative deformations, and (co)associative bialgebras
- Shoikhet
(Show Context)
Citation Context ...of the Boardman-Vogt W-construction, will be the subject of [6]. A completely different approach to bialgebras and resolutions of objects governing them can be found in a series of papers by Shoikhet =-=[20, 21, 22]-=-, and also in a recent draft by Saneblidze and Umble [19]. A general theory of resolutions of props is, besides [15], also the subject of Vallette’s thesis and its follow-up [24, 25]. Let us briefly s... |

6 | A diagonal on the associahedra
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Citation Context ...). Then define For example, with this notation, formula (24) for ∂( ) can be simplified to ∂( ) = ∂0( ) + = ∂0( ) + − ∆( ) − − + − ∆( ) , where ∆ is the Saneblidze-Umble diagonal in the associahedron =-=[14]-=-. The next term is ∂( ) = ∂0( ) − − + + + − − − − + − + + − − − − + + + + . Observe that the last term of the above equation is − ∆(3) ( ) ∆ (3) ( ) , where ∆ (3) (−) := (∆⊗ 1)∆(−) denotes the iterati... |

6 |
Dualité des Koszul des PROPs
- Vallette
- 2003
(Show Context)
Citation Context ...s by Shoikhet [20, 21, 22], and also in a recent draft by Saneblidze and Umble [19]. A general theory of resolutions of props is, besides [15], also the subject of Vallette’s thesis and its follow-up =-=[24, 25]-=-. Let us briefly sketch the strategy of the construction of our model. Consider objects (V, µ, ∆), where µ : V ⊗ V → V is an associative multiplication as in (1), ∆ : V → V ⊗ V is a coassociative comu... |

5 | On the PROP corresponding to bialgebras - Pirashvili |

5 | The Biderivative and A∞-bialgebras - Saneblidze, Umble |

4 | Homotopie rationnelle: Modèles de Chen - Tanré - 1983 |

3 |
Deformations and the coherence
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- 1993
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Citation Context ...n σ(2, 2) will be explained in Section 2). We suppose that V , as well as all other algebraic objects in this paper, are defined over a field k of characteristic zero. Let B be the k-linear prop (see =-=[6, 7]-=- for the terminology) describing bialgebras. The goal of this paper is to describe the minimal model of B, that is, a differential graded (dg) k-linear prop (M, ∂) together with a homology isomorphism... |

3 | A concept of 2 3PROP and deformation theory of (co) associative coalgebras
- Shoikhet
- 2003
(Show Context)
Citation Context ...of the Boardman-Vogt W-construction, will be the subject of [6]. A completely different approach to bialgebras and resolutions of objects governing them can be found in a series of papers by Shoikhet =-=[20, 21, 22]-=-, and also in a recent draft by Saneblidze and Umble [19]. A general theory of resolutions of props is, besides [15], also the subject of Vallette’s thesis and its follow-up [24, 25]. Let us briefly s... |

2 |
An e-mail message to M
- Kontsevich
- 2002
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Citation Context ...ist over much smaller objects tremely huge objects, difficult to work with, but 1 2 than props. These smaller objects, which we call 1 2 props, were introduced in an e-mail message from M. Kontsevich =-=[4]-=- who called them small props. The concept of 1 2props B easy. We thus proceed in two steps. makes the construction of the minimal model of 1 2 Step 1. We construct the minimal model (Γ(Ξ), ∂0) of the ... |

2 | Deformation theory of bialgebras, Hopf algebras and tensor categories - Kontsevich, Soibelman - 2002 |

2 |
cellular W-construction and products of A∞algebras
- Associahedra
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Citation Context ... + . Observe that the last term of the above equation is − ∆(3) ( ) ∆ (3) ( ) , where ∆ (3) (−) := (∆⊗ 1)∆(−) denotes the iteration of the Saneblidze-Umble diagonal which is coassociative on and (see =-=[13]-=-). The corresponding 3-dimensional polyhedron B 3 3 is shown in Figure 2.18 M. MARKL • • • • • • • • • • • • • • corona ornatus • • • • • • • • • • • • • • • • ad astra portatur • • • • • • • • • • •... |

2 |
Diagonals on the permutohedra, multiplihedra and associahedra
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- 2004
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Citation Context ... ∗). Then define For example, with this notation the formula for ∂( ) can be simplified to ∂( ) = ∂0( ) + = ∂0( ) + − ∆( ) − − + − ∆( ) , where ∆ is the Saneblidze-Umble diagonal in the associahedron =-=[18]-=-.MINIMAL MODEL OF THE PROP FOR BIALGEBRAS 17 •❍ ❍❍❍❍❍ •✟ • ✟✟✟✟✟ • ❅ ❅❅ • • � � •� Figure 1. Heptagon B 2 3. The next term is ∂( ) = ∂0( ) − − + + + − − − − + − + + − − − − + + + + . Observe that the... |

2 |
The biderivative, matrons and A∞-bialgebras
- Saneblidze, Umble
- 2004
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Citation Context ...A completely different approach to bialgebras and resolutions of objects governing them can be found in a series of papers by Shoikhet [20, 21, 22], and also in a recent draft by Saneblidze and Umble =-=[19]-=-. A general theory of resolutions of props is, besides [15], also the subject of Vallette’s thesis and its follow-up [24, 25]. Let us briefly sketch the strategy of the construction of our model. Cons... |

2 | An explicit deformation theory of (co)associative bialgebras
- Shoikhet
- 2003
(Show Context)
Citation Context ...of the Boardman-Vogt W-construction, will be the subject of [6]. A completely different approach to bialgebras and resolutions of objects governing them can be found in a series of papers by Shoikhet =-=[20, 21, 22]-=-, and also in a recent draft by Saneblidze and Umble [19]. A general theory of resolutions of props is, besides [15], also the subject of Vallette’s thesis and its follow-up [24, 25]. Let us briefly s... |

1 |
A product of A∞-algebras
- Markl, Shnider
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Citation Context ... + . Observe that the last term of the above equation is − ∆(3) ( ) ∆ (3) ( ) , where ∆ (3) (−) := (∆⊗ 1)∆(−) denotes the iteration of the Saneblidze-Umble diagonal which is coassociative on and (see =-=[10]-=-). The corresponding 3-dimensional polyhedron B 3 3 is shown in Figure 2.16 M. MARKL • • • • • • • • • • • • • • corona ornatus • • • • • • • • • • • • • • • • ad astra portatur • • • • • • • • • • •... |

1 | The biderivative and A∞-Hopf algebras. Work in Progress. E-mail address: markl@math.cas.cz - Saneblidze, Umble |