Coisotropic deformations of associative algebras and dispersionless integrable hierarchies

Abstract

The paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies, like the dispersionless KP and modified KP hierarchies and the universal Whitham's hierarchy of genus zero. It stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras. It discusses the link between the structure constants and the Hirota's tau function, and shows that the dispersionless Hirota's bilinear equations are, within this approach, a way of writing the associativity conditions for the structure constants in terms of the tau function. It also suggests a simple interpretation of the algebro-geometric construction of the universal Whitham's equations of genus zero due to Krichever.

arXiv commentminor misprints corrected

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Cited by in corpus (25)

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• Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy
• The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behavior, implicit solutions and wave breaking
• The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions
• Brackets
• Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
• Dispersionless integrable equations as coisotropic deformations. Extensions and reductions
• Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems
• Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields
• Quantum deformations of associative algebras and integrable systems
• Oriented Associativity Equations and Symmetry Consistent Conjugate Curvilinear Coordinate Nets
• Infinite hierarchies of nonlocal symmetries of the Chen--Kontsevich--Schwarz type for the oriented associativity equations
• Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System
• Dispersionful analogue of the Whitham hierarchy
• On Classification of Integrable Davey-Stewartson Type Equations
• Classical $r$-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics
• Bi-Hamiltonian structure of the Oriented Associativity Equation
• Algebraic varieties in Birkhoff strata of the Grassmannian Gr${}^{(2)}$: Harrison cohomology and integrable systems
• Coisotropic deformations of algebraic varieties and integrable systems
• Discrete integrable systems and deformations of associative algebras
• Projective differential geometry of multidimensional dispersionless integrable hierarchies
• Continuous-discrete integrable equations and Darboux transformations as deformations of associative algebras
• Cohomological, Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of Sato Grassmannian
• Non-degenerate solutions of universal Whitham hierarchy
• Menelaus relation and Fay's trisecant formula are associativity equations