## Homological Reduction Of Constrained Poisson Algebras (1997)

Venue: | J. Diff. Geom |

Citations: | 18 - 3 self |

### BibTeX

@ARTICLE{Stasheff97homologicalreduction,

author = {Jim Stasheff},

title = {Homological Reduction Of Constrained Poisson Algebras},

journal = {J. Diff. Geom},

year = {1997},

volume = {45},

pages = {221--240}

}

### OpenURL

### Abstract

This paper includes the mathematical version of the physics in [FHST] 2 Jim Stasheff

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Citation Context ...metry and/or constraints has a long history. There are several reduction procedures, all of which agree in "nice" cases [AGJ]. Some have a geometric emphasis - reducing a (symplectic) space =-=of states [MW]-=-, while others are algebraic - reducing a (Poisson) algebra of observables [SW]. Some start with a momentummap whose components are constraint functions [GIMMSY]; some start with a gauge (symmetry) al... |

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Citation Context ..., the quotient V := V=F may not be a manifold, in fact, may not even be Hausdorff. (An intermediate situation of considerable interest occurs with the quotient V=F being a stratified symplectic space =-=[LS]-=-.) Homological Reduction of Constrained Poisson Algebras 5 When (W; !) is a symplectic manifold with a smooth coisotropic submanifold, one of the nicest cases is called `regular', namely when the quot... |

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Citation Context ...of the Cartan-Chevalley-Eilenberg differential as they occur in physics are usually referred to as BRST operators. This honors seminal work of Becchi, Rouet and Stora [BRS] and, independently, Tyutin =-=[Ty]-=-. Apparently it was the search for such an operator in aid of quantization which motivated the work of Batalin, Fradkin and Vilkovisky. It was Browning and McMullan [BM] who first identified the Koszu... |

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Citation Context ...and McMullan as the Koszul complex of a regular ideal of constraints. I was able to put the FBV construction into the context of homological perturbation theory [S1] and, together with Henneaux et al =-=[FHST]-=-, extend the construction to the case of non-regular geometric constraints of first class. Independently, using a mixture of homological and C 1 -patching techniques, Dubois-Violette extended the cons... |

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Citation Context ...d principal G-bundle G ! P ! B. 4 Jim Stasheff The reduced phase space is T (A=G) where G is the group of "gauge transformations", the vertical automorphisms of P . In considering what the p=-=hysicists [BF],[BV1-3],[-=-FF], [FV],[H],[BM] did in some special cases, I recognized a homological "model" for\Omega\Gamma V; F) in roughly the sense of rational homotopy theory [Su]. This is the same sense in which ... |

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Citation Context ...and Study Leave from the University of North Carolina-Chapel Hill. Announced in the Bulletin of the American Mathematical Society as "Constrained Poisson algebras and strong homotopy representati=-=ons" [S2]-=-. This paper includes the mathematical version of the physics in [FHST] 2 Jim Stasheff 1. Preliminaries This research touches on questions which it is hoped will be of interest to mathematical physici... |

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Citation Context ... [SW] have defined an algebraic reduction in the context of group actions and momentum maps which is guaranteed to produce a reduced Poisson algebra but not necessarily a reduced space of states (cf. =-=[W2]-=-). (In contrast, Kostant and Sternberg use the Marsden-Weinstein reduction [MW].) The S-W (Sniatycki and Weinstein) reduced Poisson algebra is (C 1 (W )=I) G where V = J \Gamma1 (0) for some equivaria... |

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Citation Context ... B. 4 Jim Stasheff The reduced phase space is T (A=G) where G is the group of "gauge transformations", the vertical automorphisms of P . In considering what the physicists [BF],[BV1-3],[FF],=-= [FV],[H],[BM] did in so-=-me special cases, I recognized a homological "model" for\Omega\Gamma V; F) in roughly the sense of rational homotopy theory [Su]. This is the same sense in which the Cartan-Chevalley-Eilenbe... |

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Citation Context ...leaf dimension: The usual exterior derivative of differential forms restricts to determine the vertical exterior derivative because F is involutive. This complex is familiar in foliation theory, c.f. =-=[HH]-=-. The classical BRST-BFV construction has, in the nice cases, the same cohomology as this complex of longitudinal forms. A major motivating example for the BFV construction was provided by gauge theor... |

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Citation Context ... complex, later identified by Browning and McMullan as the Koszul complex of a regular ideal of constraints. I was able to put the FBV construction into the context of homological perturbation theory =-=[S1]-=- and, together with Henneaux et al [FHST], extend the construction to the case of non-regular geometric constraints of first class. Independently, using a mixture of homological and C 1 -patching tech... |

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Citation Context ...ctoring it as ffis : \Phi ! s\Phi ! I: (In terms of representatives ae 2 \Phi; ffiae is s \Gamma1 ae.) In other words, P\Omega s\Phi is the Koszul complex for the ideal I in the commutative algebra P =-=[Ko]-=-, [Bo]. If I is what is now known as a regular (at one time: Borel) ideal (an algebraic condition, but implied by I being the defining ideal in C 1 (W ) for V = J \Gamma1 (0) when 0 is a regular value... |

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Citation Context ...ures, all of which agree in "nice" cases [AGJ]. Some have a geometric emphasis - reducing a (symplectic) space of states [MW], while others are algebraic - reducing a (Poisson) algebra of ob=-=servables [SW]-=-. Some start with a momentummap whose components are constraint functions [GIMMSY]; some start with a gauge (symmetry) algebra whose generators, regarded as vector fields, correspond via the symplecti... |

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Citation Context ...constraints of first class. Independently, using a mixture of homological and C 1 -patching techniques, Dubois-Violette extended the construction to regular but not-necessarily-firstclass constraints =-=[D-V]-=-. I am grateful to all of the above for their input and inspiration, whether in their papers or in conversation. The present version has also profitted from conversations at the MSRI Workshop on Sympl... |

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Citation Context ... to g=h but the span of the constraints would be isomorphic to g. Here choose a splitting \Xi \Phi \Upsilon such that \Upsilon = h and \Xi �� g=h; then proceed as in the redundant case. In [FHST] =-=and [HT], the sett-=-ing is specifically that of a symplectic manifold (phase space) with a constraint submanifold ("surface") and moreover the assumption is made that locally the constraints can be separated in... |

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Citation Context ...my thanks to the referee who has read several versions with extreme care, suggesting extensive improvements, both factual and stylistic. While early revision was in progress, Kimura sent me a copy of =-=[Ki]-=- which has also had a significant influence on the present exposition, as has his continued interaction while with me at UNC as an NSF Post-Doc. Research supported in part by NSF grants DMS-8506637, D... |

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Citation Context ... - reducing a (symplectic) space of states [MW], while others are algebraic - reducing a (Poisson) algebra of observables [SW]. Some start with a momentummap whose components are constraint functions =-=[GIMMSY]-=-; some start with a gauge (symmetry) algebra whose generators, regarded as vector fields, correspond via the symplectic structure to constraints [D]. The relation between symmetry and constraints is p... |

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