Duke Mathematical Journal
- Duke Math. J.
- Volume 111, Number 3 (2002), 535-560.
A Darboux theorem for Hamiltonian operators in the formal calculus of variations
Abstract
We prove a formal Darboux-type theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context.
We include an exposition of the formal deformation theory of differential graded Lie algebras $\mathfrak {g}$ concentrated in degrees $[-1,\infty)$; the formal deformations of $\mathfrak {g}$ are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of $\mathfrak {g}$, and quasi-isomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.
Article information
Source
Duke Math. J., Volume 111, Number 3 (2002), 535-560.
Dates
First available in Project Euclid: 18 June 2004
Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087575085
Digital Object Identifier
doi:10.1215/S0012-7094-02-11136-3
Mathematical Reviews number (MathSciNet)
MR1885831
Zentralblatt MATH identifier
1100.32008
Subjects
Primary: 32G34: Moduli and deformations for ordinary differential equations (e.g. Knizhnik-Zamolodchikov equation) [See also 34Mxx]
Secondary: 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 55P62: Rational homotopy theory
Citation
Getzler, Ezra. A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111 (2002), no. 3, 535--560. doi:10.1215/S0012-7094-02-11136-3. https://projecteuclid.org/euclid.dmj/1087575085
