## PROP profile of deformation quantization (2004)

Citations: | 4 - 0 self |

### BibTeX

@TECHREPORT{Merkulov04propprofile,

author = {S. A. Merkulov},

title = {PROP profile of deformation quantization},

institution = {},

year = {2004}

}

### OpenURL

### Abstract

Using language of dg PROPs we give a new proof of existence of star products on (formal) germs of Poisson manifolds. 1.1. Theorem on quantization of Poisson structures is one of the culminating points of the deformation quantization programme initiated by F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowics, and D. Sternheimer [BFFLS]. It was first established by Kontsevich in the transcendental work [K1] as a corollary to his formality theorem. Another proof of the formality theorem was

### Citations

568 |
Deformation Quantization of Poisson Manifolds
- Kontsevich
(Show Context)
Citation Context ... of the deformation quantization programme initiated by F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowics, and D. Sternheimer [BFFLS]. It was first established by Kontsevich in the transcendental work =-=[K1]-=- as a corollary to his formality theorem. Another proof of the formality theorem was given by Tamarkin [Ta]. This paper offers a new proof of the theorem on deformation quantization of Poisson structu... |

96 |
Deformation theory and quantization. I. Deformations of symplectic structures
- Bayen, Flato, et al.
- 1978
(Show Context)
Citation Context ...orem on quantization of Poisson structures is one of the culminating points of the deformation quantization programme initiated by F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowics, and D. Sternheimer =-=[BFFLS]-=-. It was first established by Kontsevich in the transcendental work [K1] as a corollary to his formality theorem. Another proof of the formality theorem was given by Tamarkin [Ta]. This paper offers a... |

61 | Another proof of M. Kontsevich formality theorem
- Tamarkin
(Show Context)
Citation Context ...nd D. Sternheimer [BFFLS]. It was first established by Kontsevich in the transcendental work [K1] as a corollary to his formality theorem. Another proof of the formality theorem was given by Tamarkin =-=[Ta]-=-. This paper offers a new proof of the theorem on deformation quantization of Poisson structures (in fact, generic Maurer-Cartan elements in the Lie algebra of polyvector fields) using ideas of dg PRO... |

44 |
Homotopy algebras are homotopy algebras
- Markl
- 2004
(Show Context)
Citation Context ...here for the free product of PROPs, and that DefQ = lim s→∞ L ⋆ P(P<s). Clearly, (L ⋆ P(P<s),δ) is a dg sub-PROP of (DefQ,δ) for every s. We construct morphism F by an induction on s (cf. Lemma 20 in =-=[Ma2]-=-). Set F1 : L ⋆ P(P<1) = L −→ Lie � Bi∞ to be i. Assume Fs : L ⋆ P(P<s) −→ Lie � Bi∞ 5 It is precisely the next two formulae which one can take as a definition of the notion of cofibration of dg PROPs... |

28 | Koszul duality for dioperads
- Gan
(Show Context)
Citation Context ...et of compositions, { i◦j : E(m1,n1) ⊗ E(m2,n2) −→ E(m1 + m2 − 1,n1 + n2 − 1)} 1≤i≤n1 , 1≤j≤m2 which satisfy the axioms imitating the properties of the compositions i◦j in a generic PROP. We refer to =-=[G]-=-, where this notion was introduced, for a detailed list of these axioms. The free dioperad generated by an S-bimodule E is given by, D〈E〉(m,n) := ⊕ G〈E〉 G∈T(m,n) where T(m,n) is a subset of G(m,n) con... |

26 | Distributive laws and Koszulness - Markl - 1996 |

18 | A resolution (minimal model) of the PROP for bialgebras
- Markl
(Show Context)
Citation Context ... corolla from P〈E〉 can contain decorated graphs of arbitrary large genus, δ = ∑ ∞ g=0 δg. In fact the above genus completion of a free PROP is a particular example of the general completion procedure =-=[Ma3]-=-, Ê := lim ←− E/(E) n , which makes sense for any PROP E which is equipped with a morphism, ε : A → End〈k〉, to the trivial PROP. Here E := ker ε. Such a PROP is called augmented. Representation of a (... |

18 |
PROPped up graph cohomology
- Markl, Voronov
- 2003
(Show Context)
Citation Context ...th one internal vertex, n input legs and m output legs is called the (m,n)-corolla. We set G := ⊔m,nG(m,n). The free PROP, P〈E〉, generated by an S-module, E = {E(m,n)}m,n≥0, is defined by (see, e.g., =-=[MaVo]-=-) P〈E〉(m,n) := ⊕ ⎛ ⎝ ⊗ ⎞ E(Out(v),In(v)) ⎠ where G∈G(m,n) v∈v(G) 6 AutG• E(Out(v),In(v)) := Bij([m],Out(v)) ×Sm E(m,n) ×Sn Bij(In(v),[n]) with Bij standing for the set of bijections, • Aut(G) stands ... |

10 | Nijenhuis infinity and contractible dg manifolds
- Merkulov
- 2005
(Show Context)
Citation Context ...ted, in a local coordinate system, by polydifferential operators and natural contractions between duals. 15The motivating examples are ∧ • TM, DM and the sheaf of Nijenhuis dg Lie algebras on M (see =-=[Me2]-=-). By assumption (i), a choice of a local coordinate system on M, identifies GM with a subspace in ⊕ ˆ⊙ • M ∗ ⊗ Hom(M ⊗p ,M ⊗m ) = ∏ Hom(⊙ p M ⊗ M ⊗q ,M ⊗m ) ⊂ ∏ Hom(M ⊗n M ⊗m ). p,m≥0 p,q,m≥0 m,n≥0 L... |

5 |
Prop profile of Poisson geometry, preprint math.DG/0401034
- Merkulov
(Show Context)
Citation Context ...ssociated to the species of Poisson structures and, respectively, species of star products. The main ingredients of our proof are (i) a surprisingly small PROPeradic code of Poisson geometry found in =-=[Me1]-=-, (ii) deformation quantization of Lie 1-bialgebras and (iii) an observation that the adjoint, ✷, to the forgetful functor PROP → 1 2PROP (see [K2, MaVo]) as well as its further PROP → PROP introduced... |

4 |
letter to Martin Markl
- Kontsevich
- 2002
(Show Context)
Citation Context ... of G(m,n) consisting of connected trees (i.e., connected graphs of genus 0). Another less obvious (and, probably, much more important) reduction of the notion of PROP was introduced by Kontsevich in =-=[K2]-=- and studied in detail in [MaVo]: A 1 2PROP is an , equipped with two sets of compositions, S-bimodule, E = {E(m,n)} m,n≥1 m+n≥3 and { 1◦j : E(m1,1) ⊗ E(m2,n2) −→ E(m1 + m2 − 1,n2)} 1≤j≤m2 { i◦1 : E(m... |