dorsal/arxiv
View SchemaHomological Reduction of Constrained Poisson Algebras
| Authors | Jim Stasheff |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9603021 |
| URL | https://arxiv.org/abs/q-alg/9603021 |
Abstract
The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson manifold $W$ by the ideal $I$ of functions which vanish on a constraint locus. This ideal is called first class if $I$ is closed under the Poisson bracket; geometers refer to the constraint locus as coisotropic. The physicists' model is crucially a differential Poisson algebra extension of a Poisson algebra $P$; its differential contains a piece which reinvented the Koszul complex for the ideal $I$ and a piece which looks like the Cartan-Chevalley-Eilenberg differential. The present paper is concerned purely with the homological (Poisson) algebraic structures, using the notion of ``model'' from rational homotopy theory and the techniques of homological perturbation theory to establish some of the basic results explaining the mathematical existence of the classical BRST-BFV construction. Although the usual treatment of BFV is basis dependent (individual constraints) and nominally finite dimensional, I take care to avoid assumptions of finite dimensionality and work more invariantly in terms of the ideal. In particular, the techniques are applied to the `irregular' case (the ideal is not generated by a regular sequence of constraints), although the geometric interpretation is less complete.
{
"annotation_id": "16739ccd-8e8a-45c2-b48d-c91c9836a6e7",
"date_created": "2026-03-02T18:01:28.137000Z",
"date_modified": "2026-03-02T18:01:28.137000Z",
"file_hash": "4513830f7918052c851826fb020130fb90b074b3a95e314fdd4e8ec83da58e8d",
"private": false,
"record": {
"abstract": "The ``classical BRST construction\u0027\u0027 as developed by\nBatalin-Fradkin-Vilkovisky is a homological construction for the reduction of\nthe Poisson algebra $P = C^\\infty (W)$ of smooth functions on a Poisson\nmanifold $W$ by the ideal $I$ of functions which vanish on a constraint locus.\nThis ideal is called first class if $I$ is closed under the Poisson bracket;\ngeometers refer to the constraint locus as coisotropic. The physicists\u0027 model\nis crucially a differential Poisson algebra extension of a Poisson algebra $P$;\nits differential contains a piece which reinvented the Koszul complex for the\nideal $I$ and a piece which looks like the Cartan-Chevalley-Eilenberg\ndifferential.\n The present paper is concerned purely with the homological (Poisson)\nalgebraic structures, using the notion of ``model\u0027\u0027 from rational homotopy\ntheory and the techniques of homological perturbation theory to establish some\nof the basic results explaining the mathematical existence of the classical\nBRST-BFV construction. Although the usual treatment of BFV is basis dependent\n(individual constraints) and nominally finite dimensional, I take care to avoid\nassumptions of finite dimensionality and work more invariantly in terms of the\nideal. In particular, the techniques are applied to the `irregular\u0027 case (the\nideal is not generated by a regular sequence of constraints), although the\ngeometric interpretation is less complete.",
"arxiv_id": "q-alg/9603021",
"authors": [
"Jim Stasheff"
],
"categories": [
"q-alg",
"dg-ga",
"hep-th",
"math.DG",
"math.QA"
],
"title": "Homological Reduction of Constrained Poisson Algebras",
"url": "https://arxiv.org/abs/q-alg/9603021"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "7f5a1185-0241-4686-b611-fe144c203913",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}