A Darboux theorem for Hamiltonian operators in the formal calculus of variations

By E. Getzler. 2000

156 citations recorded by the source corpus.

We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes three main concepts: 1) dg Lie algebras concentrated in degrees [-1,\infty) such as the Schouten algebra - these give a convenient language for describing deformation problems; 2) the Deligne 2-groupoid associated to such a dg Lie algebra, which represents the moduli of formal deformations; 3) a refined version of the Schouten bracket in the formal calculus of variations, due to V. O. Soloviev (hep-th/9305133).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7c92548999a64b5d118076942491d18c0b9d19ae

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In Algebra and Number Theory

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Citations in the corpus

Deformation quantization of gerbes (2005, background): “[15,16] and references thereof) of a cosimplicial DGLA and prove that the set of isomorphism classes of such data does not change if one passes to a quasi-isomorphic cosimplicial DGLA (Proposition 3.”
On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour (2005, background): “Using results of [15] (see also [6, 9]) one can show that any Hamiltonian perturbation of the equation (1.”
Geometry of jet spaces and integrable systems (2010, background): “The first cohomology group of ∂A is trivial (see [42]), and consequently we obtain an infinite series of first integrals and the corresponding symmetries.”
On weak Lie 2‐algebras (2007, background): “It was Getzler [2] who generalized this construction to L∞-algebras and higher values of n.”