On the Quantum Deformations of Associative Sato Grassmannian Algebras and the Related Matrix Problems

Alexander A. Balinsky, Victor A. Bovdi, Anatolij K. Prykarpatski

Research output: Contribution to journalArticlepeer-review


We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix problems made it possible, in part, to describe some kinds of Frobenius manifolds within the Dubrovin-type reformulation of the well-known WDVV associativity equations, previously derived in topological field theory. In particular, we state that these equations are equivalent to some bi-Hamiltonian flows on a smooth functional submanifold with respect to two compatible Poisson structures, generating a countable hierarchy of commuting to each other’s hydrodynamic flows. We also studied the inverse problem aspects of the quantum Grassmannian deformation Lie algebraic structures, related with the well-known countable hierarchy of the higher nonlinear Schrödinger-type completely integrable evolution flows.

Original languageEnglish
Article number54
Issue number1
Publication statusPublished - Jan 2024


  • Casimir invariants
  • Lax integrability
  • Lax–Sato equations
  • Lie algebraic scheme
  • Lie–Poisson structure
  • Sato Grassmannians
  • associativity
  • co-adjoint action
  • heavenly equations
  • loop Lie algebra
  • torus diffeomorphisms

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • General Mathematics
  • Physics and Astronomy (miscellaneous)


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