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**The (secret?) homological algebra of the Batalin-Vilkovisky approach.**
*(English)*
Zbl 0969.17012

Henneaux, Marc (ed.) et al., Secondary calculus and cohomological physics. Proceedings of a conference, Moscow, Russia, August 24-31, 1997. Providence, RI: AMS, American Mathematical Society. Contemp. Math. 219, 195-210 (1998).

An old observation of E. Noether says that a non-regular Lagrangian induces non-trivial identities among the solutions of the Euler-Lagrange equations (and their derivatives) in Lagrangian field theory, and these identities, in turn, give rise to gauge symmetries [E. Noether, Gött. Nachr. 1918, 235-257 (1918; JFM 46.0770.01)]. On the infinitesimal level, these symmetries do not arise via ordinary Lie algebra actions on the fields. Incorporating ghosts led eventually to the Batalin-Vilkovisky approach, on the classical as well as the quantum level. The author had already noticed that sh-Lie algebras and homological algebra are lurking behind the Batalin-Vilkovisky approach to classical Lagrangian field theory.

The purpose of the paper is to review this homological algebra approach which materializes via a combination of the BV machinery with the variational bicomplex. The classical version of the master equation then has the form of the integrability condition of deformation theory, and the Noether identities may be seen as syzygies. The ghosts appear as Koszul-Tate generators of a resolution. The corresponding Hamiltonian picture may be found in the author’s paper [J. Differ. Geom. 45, No. 1, 221-240 (1997; Zbl 0874.58020)].

For the entire collection see [Zbl 0895.00065].

The purpose of the paper is to review this homological algebra approach which materializes via a combination of the BV machinery with the variational bicomplex. The classical version of the master equation then has the form of the integrability condition of deformation theory, and the Noether identities may be seen as syzygies. The ghosts appear as Koszul-Tate generators of a resolution. The corresponding Hamiltonian picture may be found in the author’s paper [J. Differ. Geom. 45, No. 1, 221-240 (1997; Zbl 0874.58020)].

For the entire collection see [Zbl 0895.00065].

Reviewer: Johannes Huebschmann (Villeneuve d’Ascq)

### MSC:

17B55 | Homological methods in Lie (super)algebras |

16E45 | Differential graded algebras and applications (associative algebraic aspects) |

81T70 | Quantization in field theory; cohomological methods |

37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |

18G55 | Nonabelian homotopical algebra (MSC2010) |

70H03 | Lagrange’s equations |

70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |

70H45 | Constrained dynamics, Dirac’s theory of constraints |

70H50 | Higher-order theories for problems in Hamiltonian and Lagrangian mechanics |