TY - JOUR

T1 - A darboux theorem for hamiltonian operators in the formal calculus of variations

AU - Getzler, Ezra

PY - 2002

Y1 - 2002

N2 - 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 g concentrated in degrees [−1, ∞); the formal deformations of g are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of g, and quasiisomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.

AB - 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 g concentrated in degrees [−1, ∞); the formal deformations of g are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of g, and quasiisomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.

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U2 - 10.1215/S0012-7094-02-11136-3

DO - 10.1215/S0012-7094-02-11136-3

M3 - Article

AN - SCOPUS:0037085530

VL - 111

SP - 535

EP - 560

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 3

ER -