## On the structure of cofree Hopf algebras

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Venue: | J. reine angew. Math |

Citations: | 33 - 4 self |

### BibTeX

@ARTICLE{Loday_onthe,

author = {Jean-louis Loday and María Ronco},

title = {On the structure of cofree Hopf algebras},

journal = {J. reine angew. Math},

year = {},

pages = {123--155}

}

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### Abstract

Abstract. We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B∞-algebra. We construct a universal enveloping functor U2 from nondifferential B∞-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the description of the free 2as-algebra in terms of planar trees to unravel the operad associated to nondifferential B∞-algebras.

### Citations

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(Show Context)
Citation Context ... U(Prim H), where the primitive part Prim H is viewed as a Lie algebra, and U is the universal enveloping functor. This result is known as the Cartier-Milnor-Moore theorem in the literature, cf. [4], =-=[16]-=- and [18] appendix B. Combined with the Poincaré-Birkhoff-Witt theorem, it gives an equivalence between the cofree cocommutative Hopf algebras and the Hopf algebras of the form U(g), where g is a Lie ... |

230 |
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Citation Context ...x) 2 − (1 + x)C(x) + 1 = 0. We deduce from itsON THE STRUCTURE OF COFREE HOPF ALGEBRAS 21 the expression given in 5.1: � Cnx n = (1 + x − � 1 − 6x + x2 )/4x . n≥0 5.9. Homology and Koszul duality. In =-=[8]-=- Ginzburg and Kapranov developed the theory of Koszul duality for binary quadratic operads. When an operad is Koszul, its Koszul dual permits us to construct a small chain complex to compute the homol... |

227 |
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Citation Context ...ductive relation aω ∗ bθ = M11(a, b)(ω ∗ θ) + a(ω ∗ bθ) + b(aω ∗ θ) , where a, b ∈ R and ω, θ are tensors. (c) If Mpq = 0 for all (p, q) such that p ≥ 2, then we get a brace algebra, cf. for instance =-=[6]-=-, [11], [20], [21]. (d) A prop is a family of Sn × Sop m -modules P(n, m) equipped with a composition P(n1, m1)⊗· · ·⊗ P(np, mp)⊗P(r1, s1)⊗· · ·⊗ P(rq, sq) γ → P(n1 + · · ·np, s1 + · · · sq) for m1 + ... |

225 |
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(Show Context)
Citation Context ...), where the primitive part Prim H is viewed as a Lie algebra, and U is the universal enveloping functor. This result is known as the Cartier-Milnor-Moore theorem in the literature, cf. [4], [16] and =-=[18]-=- appendix B. Combined with the Poincaré-Birkhoff-Witt theorem, it gives an equivalence between the cofree cocommutative Hopf algebras and the Hopf algebras of the form U(g), where g is a Lie algebra. ... |

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la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts (French
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Citation Context ...imilar context in [19]. Observe that in our case the characteristic zero hypothesis is not needed since the geometric series has no denominators. Theorem 2.6 is similar to the Hopf-Borel theorem, cf. =-=[3]-=-, which states, in the non-graded case, that any connected commutative cocommutative Hopf algebra H is isomorphic to the symmetric algebra S(Prim H) (in characteristic zero). 3. 2-associative algebra ... |

124 |
homotopy algebra and iterated integrals for double loop spaces arXiv:hep-th/9403055v1
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Citation Context ... a Lie algebra. A B∞-algebra is defined by (p + q)-ary operations for any pair of positive integers (p, q) satisfying some relations. This structure appears naturally in algebraic topology. (cf. [2], =-=[7]-=-, [11], [21]). The universal enveloping functor U is replaced by a functor U2 from B∞-algebras to 2-associative algebras, which are vector spaces equipped with two associative operations sharing the s... |

120 |
Hopf algebra of the planar binary trees
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Citation Context ...∞ to dendriform algebras. 7.8. Comparison of Hopf algebras of trees. As an associative algebra the free dendriform algebra on one generator Dend(K) is a tensor algebra on the planar binary trees (cf. =-=[14]-=-). The Connes-Kreimer Hopf algebra HCK is the symmetric algebra on (non-planar) rooted trees. Forgetting planarity and symmetrizing gives a surjection of Hopf algebras Dend(K) ։ HCK (cf. for instance ... |

118 | Hopf algebras, renormalization and noncommutative geometry
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Citation Context ...scribe the other product · and the coproduct ∆ in terms of trees. This description is very similar to the description of the Hopf algebra of (non-planar) rooted trees given by Connes and Kreimer (cf. =-=[4]-=-). Here the rooted trees are replaced by the planar rooted trees, and the polynomials by the non-commutative polynomials. The free Lie algebra is a complicated object which is mainly studied through i... |

99 |
algebras, renormalization and noncommutative
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Citation Context ...scribe the other product · and the coproduct ∆ in terms of trees. This description is very similar to the description of the Hopf algebra of (non-planar) rooted trees given by Connes and Kreimer (cf. =-=[5]-=-). Here the rooted trees are replaced by the planar rooted trees, and the polynomials by the non-commutative polynomials. The free Lie algebra is a complicated object which is mainly studied through i... |

74 | The cohomology structure of an associative - Gerstenhaber - 1963 |

61 |
Coalgebras and bialgebras in combinatorics
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Citation Context ...The map ∆ defined by ∆(xn ) = �n p=0 xp ⊗ xn−p satisfies the unital infinitesimal relation. The unital infinitesimal relation differs from the infinitesimal relation used by S. Joni and G.-C. Rota in =-=[10]-=- (see also [1]) by the presence of the term −x ⊗ y. From our relation it comes ∆(1) = 1 ⊗ 1. Recall that the notion of connectedness given in 1.2 uses only ∆ and 1, and so is applicable here. 2.3. Pro... |

53 |
Dialgebras and Related Operads
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Citation Context ...→ A where b ′ is given by b ′ (a1, · · · , an) = � i=n−1 i=1 (−1) i (a1, · · · , aiai−1, · · · , an). Its homology is denoted H As ∗ (A). From the definition of the dual of an operad (cf. loc.cit. or =-=[12]-=- Appendix B, for a brief introduction), the dual of the operad 2as is the operad 2as ! whose algebras have two associative operations ∗ and · verifying: (x ∗ y) · z = 0 = x ∗ (y · z) (x · y) ∗ z = 0 =... |

29 |
Scindement d’associativité et algèbres de Hopf. Actes des journées mathématiques à la mémoire de
- Loday
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(Show Context)
Citation Context ...ke the product ∗ into a product in T sh (V )[[h]] by taking the element in V ⊗p+q−i as coefficient of h i .sON THE STRUCTURE OF COFREE HOPF ALGEBRAS 7 2. Unital infinitesimal bialgebra We recall from =-=[13]-=- the notion of unital infinitesimal bialgebra and we prove a structure theorem. 2.1. Definition. A unital infinitesimal bialgebra (H, ·, ∆) is a vector space H equipped with a unital associative produ... |

27 |
Eulerian idempotents and Milnor-Moore theorem for certeins non cocommutative Hopf algebras
- Ronco
(Show Context)
Citation Context ...ation aω ∗ bθ = M11(a, b)(ω ∗ θ) + a(ω ∗ bθ) + b(aω ∗ θ) , where a, b ∈ R and ω, θ are tensors. (c) If Mpq = 0 for all (p, q) such that p ≥ 2, then we get a brace algebra, cf. for instance [6], [11], =-=[20]-=-, [21]. (d) A prop is a family of Sn × Sop m -modules P(n, m) equipped with a composition P(n1, m1)⊗· · ·⊗ P(np, mp)⊗P(r1, s1)⊗· · ·⊗ P(rq, sq) γ → P(n1 + · · ·np, s1 + · · · sq) for m1 + · · · + mp =... |

26 | The structure of the A(∞)-algebra, and the Hochschild and Harrison cohomologies, (Russian) Trudy Tbiliss
- Kadeishvili
- 1988
(Show Context)
Citation Context ...e algebra. A B∞-algebra is defined by (p + q)-ary operations for any pair of positive integers (p, q) satisfying some relations. This structure appears naturally in algebraic topology. (cf. [2], [7], =-=[11]-=-, [21]). The universal enveloping functor U is replaced by a functor U2 from B∞-algebras to 2-associative algebras, which are vector spaces equipped with two associative operations sharing the same un... |

24 |
Comparison of Hopf algebras on trees
- Holtkamp
(Show Context)
Citation Context ...). The Connes-Kreimer Hopf algebra HCK is the symmetric algebra on (non-planar) rooted trees. Forgetting planarity and symmetrizing gives a surjection of Hopf algebras Dend(K) ։ HCK (cf. for instance =-=[9]-=-). Since a dendriform algebra is a particular case of dipterous algebra, there is a morphism of dipterous algebras (hence of Hopf algebras): Dipt(K) → Dend(K) Since Dipt(K) is a tensor algebra over th... |

23 |
Primitive elements in a free dendriform algebra, New trends in Hopf algebra theory (La Falda
- Ronco
- 1999
(Show Context)
Citation Context ...− uc. It is the analogue of the first Eulerian idempotent, which, in the classical case, is defined as the logarithm series applied to J. The geometric series was already used in a similar context in =-=[19]-=-. Observe that in our case the characteristic zero hypothesis is not needed since the geometric series has no denominators. Theorem 2.6 is similar to the Hopf-Borel theorem, cf. [3], which states, in ... |

20 |
Infinitesimal Hopf algebras, in: New Trends in Hopf Algebra Theory (La Falda
- Aguiar
- 1999
(Show Context)
Citation Context ...ed by ∆(xn ) = �n p=0 xp ⊗ xn−p satisfies the unital infinitesimal relation. The unital infinitesimal relation differs from the infinitesimal relation used by S. Joni and G.-C. Rota in [10] (see also =-=[1]-=-) by the presence of the term −x ⊗ y. From our relation it comes ∆(1) = 1 ⊗ 1. Recall that the notion of connectedness given in 1.2 uses only ∆ and 1, and so is applicable here. 2.3. Proposition-Notat... |

20 |
Algèbres de Hopf colibres
- Loday, Ronco
- 2003
(Show Context)
Citation Context ...the cofree Hopf algebras whose primitive part is in fact a brace algebra. Then, in section 9, we compare several results of the same kind. The main result of this paper was announced without proof in =-=[15]-=-. Notation. In this paper K is a field and all vector spaces are over K. Its unit is denoted 1K or just 1. The vector space spanned by the elements of a set X is denoted K[X]. The tensor product of ve... |

19 | Homotopy Gerstenhaber algebras - Voronov |

5 |
Hyperalgèbres et groupes de Lie formels, Séminaire Sophus Lie, 2e année: 1955/56. Faculté des Sciences de Paris
- Cartier
(Show Context)
Citation Context ... form U(Prim H), where the primitive part Prim H is viewed as a Lie algebra, and U is the universal enveloping functor. This result is known as the Cartier-Milnor-Moore theorem in the literature, cf. =-=[4]-=-, [16] and [18] appendix B. Combined with the Poincaré-Birkhoff-Witt theorem, it gives an equivalence between the cofree cocommutative Hopf algebras and the Hopf algebras of the form U(g), where g is ... |

2 |
The double bar and cobar
- Baues
- 1981
(Show Context)
Citation Context ...ad of a Lie algebra. A B∞-algebra is defined by (p + q)-ary operations for any pair of positive integers (p, q) satisfying some relations. This structure appears naturally in algebraic topology. (cf. =-=[2]-=-, [7], [11], [21]). The universal enveloping functor U is replaced by a functor U2 from B∞-algebras to 2-associative algebras, which are vector spaces equipped with two associative operations sharing ... |

2 |
Sets with two associative operations. Cent
- Pirashvili
(Show Context)
Citation Context ...int if ◦s = ◦ (resp. ◦t = ◦). 5.6. Remark. One can switch the roles of the two products, in particular 2as(V ) is free as an associative algebra for · . A free 2-associative set is called a duplex in =-=[17]-=-. Corollary 5.5 gives also the structure of the free duplex in one generator. Observe that for s = s1 ∨ . . . ∨ sn one has s ∗ = s · 1 ∗ . . . ∗ s · n and s · = s ∗ 1 · . . . · s ∗ n . In figure 1 we ... |

2 |
Homotopy Gerstenhaber algebras, Conf. Moshe Flato
- Voronov
- 1999
(Show Context)
Citation Context ...bra. A B∞-algebra is defined by (p + q)-ary operations for any pair of positive integers (p, q) satisfying some relations. This structure appears naturally in algebraic topology. (cf. [2], [7], [11], =-=[21]-=-). The universal enveloping functor U is replaced by a functor U2 from B∞-algebras to 2-associative algebras, which are vector spaces equipped with two associative operations sharing the same unit. We... |

1 |
Comparison of Hopf algebras on trees
- unknown authors
(Show Context)
Citation Context ...). The Connes-Kreimer Hopf algebra HCK is the symmetric algebra on (non-planar) rooted trees. Forgetting planarity and symmetrizing gives a surjection of Hopf algebras Dend(K) ։ HCK (cf. for instance =-=[8]-=-). Since a dendriform algebra is a particular case of dipterous algebra, there is a morphism of dipterous algebras (hence of Hopf algebras): Dipt(K) → Dend(K) Since Dipt(K) is a tensor algebra over th... |