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A Darboux theorem for Hamiltonian operators in the formal calculus of variations. (English) Zbl 1100.32008
The author proves 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. It is shown that the Schouten Lie algebra is a formal differential graded Lie algebra. An exposition is included of the formal deformation theory of differential graded Lie algebras concentrated in degrees $$[-1,\infty )$$.

##### MSC:
 32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures 35Q53 KdV equations (Korteweg-de Vries equations) 55P62 Rational homotopy theory
##### Keywords:
Darboux theorem; graded Lie algebra; KdV hierarchies
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##### References:
 [1] L. A. Dickey, “Poisson brackets with divergence terms in field theories: Three examples” in Higher Homotopy Structures in Topology and Mathematical Physics (Poughkeepsie, N. Y., 1996) , Contemp. Math. 227 , Amer. Math. Soc., Providence, 1999, 67–78. · Zbl 0988.37099 [2] B. A. Dubrovin and S. P. Novikov, Hydrodynamics of weakly deformed soliton lattices: Differential geometry and Hamiltonian theory , Uspekhi Mat. Nauk 44 , no. 6 (1989), 29–98., 203; English translation in Russian Math. Surveys 44 , no. 6 (1989), 35–124. · Zbl 0712.58032 [3] I. M. Gelfand and I. Ja. Dorfman, Schouten bracket and Hamiltonian operators , Funktsional. Anal. i Prilozhen. 14 (1980), 71–74. · Zbl 0444.58010 [4] W. M. Goldman and J. J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds , Inst. Hautes Études Sci. Publ. Math. 67 (1988), 43–96. · Zbl 0678.53059 [5] M. Kontsevich, Deformation quantization of Poisson manifolds, I , · Zbl 1058.53065 [6] M. D. Kruskal, R. M. Miura, C. S. Gardner, and N. J. Zabusky, Korteweg-de Vries equation and generalizations, V: Uniqueness and nonexistence of polynomial conservation laws , J. Mathematical Phys. 11 (1970), 952–960. · Zbl 0283.35022 [7] I. Moerdijk and J.-A. Svensson, Algebraic classification of equivariant homotopy 2-types, I , J. Pure Appl. Algebra 89 (1993), 187–216. · Zbl 0787.55008 [8] A. Nijenhuis and R. W. Richardson Jr., Cohomology and deformations of algebraic structures , Bull. Amer. Math. Soc. 70 (1964), 406–411. · Zbl 0138.26301 [9] P. Olver, “Hamiltonian perturbation theory and water waves” in Fluids and Plasmas: Geometry and Dynamics (Boulder, 1983) , Contemp. Math. 28 , Amer. Math. Soc., Providence, 1984, 231–249. · Zbl 0521.76018
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