## The Deformation Complex for DG Hopf Algebras (2002)

### BibTeX

@MISC{Umble02thedeformation,

author = {Ronald N. Umble},

title = {The Deformation Complex for DG Hopf Algebras},

year = {2002}

}

### OpenURL

### Abstract

Abstract. Let H be a DG Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the Hochschild-Cartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H (q)-structure. The direct limit of all such truncations is the appropriate setting for deforming H as a strongly homotopy associative structure. Sign complications are systematically controlled. The connection between rational perturbation theory and the deformation theory of certain free commutative differential graded algebras is clarified. 1.

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Citation Context ...d R-module A together with maps {µ(ℓ) ∈ Hom 2−ℓ R (A⊗ℓ, A)}1≤ℓ≤n such that for each ℓ ≤ n, ∑ 0≤i<j; j+k=ℓ+1 (−1) i+ik+ℓk+k µ (j) ◦ (1 ⊗i ⊗ µ (k) ⊗ 1 ⊗(j−i−1) ) = 0. The signs here agree with those in =-=[23]-=-; we use upper indices and reserve the lower for indexing coefficients in a deformation. An A(n)-algebra is strict if µ(n) = 0. Every d.g.a. (A, d, µ) is a strict A(n)-algebra for all n ≥ 3 via µ(1) =... |

269 |
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Citation Context ...ted and biassociative, there is a unique inductively defined antipode S that acts as the identity in degree 0 and by S(x) = −x − ∑ x (1)S(x (2)) in positive degrees, where ∆(x) = ∑ x (1) ⊗ x (2); see =-=[20]-=-. A graded R-bialgebra (H, µ, η, ∆, ε) equipped with antipode S is a graded R-Hopf algebra (g.h.a.). Furthermore, if (H, d, µ) is a d.g.a., (H, d, ∆) is a d.g.c. and (H, µ, η, ∆, ε, S) is a g.h.a., th... |

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Citation Context ...MATION COMPLEX FOR DG HOPF ALGEBRAS 13 deformation appear as an inductively defined sequence of cocycles in ˜ B3 (A; A; n). We note that the deformation theory of A as an A(∞)-algebra also appears in =-=[15]-=- and [21]. The case n = 3 is discussed in some detail in section 6 (as a special case) and subsequently by Lazarev and Movshev in the sequel [17]. When n = 3, we adopt the standard notation At = (A[[t... |

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Citation Context ...a. These resolutions, which are not meant to model chains on some contractible space, have differentials of internal degree zero and avoid the dimension shifts of Adams [1] and Eilenberg and Mac Lane =-=[8]-=-. Section 4 dualizes and generalizes the notion of a differential graded bimodule over a differential graded algebra, which is implicit in [19], to analogous structures over differential graded coalge... |

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Citation Context ...n cohomology of (Λ, d) and the homology of {Coder(L c Λ, A), d c }, where L c Λ is the free d.g. Lie coalgebra on Λ and d c is the differential induced by d, was given by Schlessinger and Stasheff in =-=[22]-=-. 5.2. The Deformation Complex for Differential Graded Coalgebras. Let (C, d, ∆) be a d.g.c. The deformation complex for C is a double complex dual to the one discussed in 5.1; this discussion is incl... |

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Citation Context ...s two-fold: (1) to give an explicit construction of the deformation complex for differential graded Hopf algebras and (2) to relate the rational perturbation theory of Felix [9] and Halperin-Stasheff =-=[11]-=- to the deformation theory of certain free commutative differential graded algebras. The untruncated deformation complex constructed here directs the deformation of a differential graded Hopf algebra ... |

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Citation Context ...section 6 below). Felix and Halperin-Stasheff apply this theory in two somewhat different ways. One can obtain a minimal model for A as the limit of a Tate-Jozefiak resolution16 RONALD N. UMBLE of A =-=[24]-=-, [13]. Thought of this way, the minimal model is naturally bigraded with respect to internal and resolution degrees. In either approach, this ”bigraded model” is perturbed to a ”filtered model”; the ... |

40 |
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Citation Context ... differential graded (co)algebra. These resolutions, which are not meant to model chains on some contractible space, have differentials of internal degree zero and avoid the dimension shifts of Adams =-=[1]-=- and Eilenberg and Mac Lane [8]. Section 4 dualizes and generalizes the notion of a differential graded bimodule over a differential graded algebra, which is implicit in [19], to analogous structures ... |

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Citation Context .... Proceed inductively until f is totally cohomologous to some (n, n)-cocycle hn. I should note that ”staircase” arguments such as this are not new, having appeared as early as 1952 in a paper by Weil =-=[25]-=-. Definition 14. Let (M, λ, ρ) be an A-bimodule. A k-linear map θ : A → M is a derivation if θ ◦µ = ρ ◦(θ ⊗1)+λ◦(1 ⊗θ). The set of all derivations of degree p is denoted by Der p (A, M). Corollary 1. ... |

21 |
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Citation Context ...itruncated Harrison cochains and ˜ Harr∗ c.d.g.a. (A; M; n) denotes the corresponding cohomology. The complex { ˜ Ch∗ (A; A; n), DB} is the deformation complex for A as a ”balanced” A(n)-algebra; see =-=[14]-=- and [19]. If (A, µ) is a c.g.a. sans differential, one can forget the internal grading and grade the Harrison cohomology externally (with respect to the number of tensor factors) as one does classica... |

18 | A∞ Algebras and the Cohomology of Moduli Spaces
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Citation Context ...ebra, A (n)-algebra, deformation. This research funded in part by a Millersville University Faculty Reasearch Grant. 12 RONALD N. UMBLE in characteristic zero. The recent work of Penkava and Schwarz =-=[21]-=- demonstrates that careful attention to signs can be critical. This paper is organized as follows: Section 2 establishes the necessary preliminaries and section 3 reviews the ”classical” (co)bar resol... |

12 |
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Citation Context ...to ”move past” the resolution differentials without complicating signs. While the first strategy is evident in Gerstenhaber and Schack’s exposition [10], the second was used by Burghelea and Poirrier =-=[3]-=- to define the Hochschild and cyclic cohomologies of free commutative associative differential graded algebras 1991 Mathematics Subject Classification. 16W30, 13D10, 16E45, 55P62. Key words and phrase... |

7 |
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Citation Context ...n 6 below). Felix and Halperin-Stasheff apply this theory in two somewhat different ways. One can obtain a minimal model for A as the limit of a Tate-Jozefiak resolution16 RONALD N. UMBLE of A [24], =-=[13]-=-. Thought of this way, the minimal model is naturally bigraded with respect to internal and resolution degrees. In either approach, this ”bigraded model” is perturbed to a ”filtered model”; the pertur... |

7 |
Homology
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(Show Context)
Citation Context ...strates that careful attention to signs can be critical. This paper is organized as follows: Section 2 establishes the necessary preliminaries and section 3 reviews the ”classical” (co)bar resolution =-=[18]-=- of a graded (co)algebra and its extension to a differential graded (co)algebra. These resolutions, which are not meant to model chains on some contractible space, have differentials of internal degre... |

7 |
A Cohomology Theory for A(m)-algebras and
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(Show Context)
Citation Context ...d Harrison cochains and ˜ Harr∗ c.d.g.a. (A; M; n) denotes the corresponding cohomology. The complex { ˜ Ch∗ (A; A; n), DB} is the deformation complex for A as a ”balanced” A(n)-algebra; see [14] and =-=[19]-=-. If (A, µ) is a c.g.a. sans differential, one can forget the internal grading and grade the Harrison cohomology externally (with respect to the number of tensor factors) as one does classically. Let ... |

5 |
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(Show Context)
Citation Context ...he purpose of this paper is two-fold: (1) to give an explicit construction of the deformation complex for differential graded Hopf algebras and (2) to relate the rational perturbation theory of Felix =-=[9]-=- and Halperin-Stasheff [11] to the deformation theory of certain free commutative differential graded algebras. The untruncated deformation complex constructed here directs the deformation of a differ... |

4 | Cohomologie des Coalgebras - Cartier |

4 |
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Citation Context ...rs) as one does classically. Let Harr∗ (A; M) denote the Harrison cohomology graded in this way; one has the following result, which is a consequence of the Hochschild, Kostant, and Rosenberg Theorem =-=[12]-=-: Theorem 1. Let k be a field of characteristic 0. If A is a free commutative k-algebra and (M, λ, µ) is any symmetric A-bimodule, then Harr n (A; M) = 0 whenever n > 1. The requirement that k have ch... |

4 |
Strongly homotopy lie algebras, Preprint hep-th 9406095
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(Show Context)
Citation Context ... model with coefficients in itself. We observe that for free c.d.g.a.’s, all flexibility lies in the direction of the differential. The Lie algebra analogs of these constructions recently appeared in =-=[16]-=-. Next we dualize and obtain the Cartier cohomology of a connected coassociative differential graded coalgebra (d.g.c.) C with coefficients in a differential graded Cbicomodule N; the deformation comp... |

2 |
Notion de Construction,” Séminaire H
- Cartan
(Show Context)
Citation Context ...spective bidegree (1, 0) and (0, −1) (see Figure 1). This resolution is acyclic with respect to ∂ via the contracting homotopy s given above. An isomorphic (and more familiar) construction appears in =-=[4]-=- but with the Eilenberg-Mac Lane shift in dimension. This dimension shift introduces a set of signs that give rise to a standard double complex whose subdiagrams anticommute; in this case D = d (∗) + ... |

2 |
O Kvazitreoogolnix KvaziHopfix Algebrax i Odnoi Groope, Tessno Svyazannoi c Gal(Q/Q
- Drinfel’d
(Show Context)
Citation Context ...icular theory to analyze the deformations of the de Rham cochains on a Lie group. Finally, we note that the appropriate setting for the deformation theory of Hopf algebras as quasi-Hopf algebras [6], =-=[7]-=- is {C 0,m,n (H; H), ∂C, δC}m≥1; n≥0, i.e., the subcomplex of cochains in the ”semi-restricted coordinate plane” p = 0, m ≥ 1.THE DEFORMATION COMPLEX FOR DG HOPF ALGEBRAS 21 I must express my sincere... |

2 |
Deformations of the de Rham Algebra
- Lazarev, Movshev
(Show Context)
Citation Context ... H (∞)-structure; appropriate truncations direct the deformation of H as an H(q)-structure. The special case q = 3 is applied by Lazarev and Movshev in their paper Deformations of the de Rham Algebra =-=[17]-=-, which follows as a sequel. In [10], Gerstenhaber and Schack showed how to deform a biassociative Hopf algebra H over a field k relative to its algebraic cohomology. Following their cues, we define t... |