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On the quantum deformations of associative Sato Grassmannian algebras and the related matrix problems

Balinsky, Alexander A. ORCID:, Bovdi, Victor A. and Prykarpatski, Anatolij K. 2024. On the quantum deformations of associative Sato Grassmannian algebras and the related matrix problems. Symmetry 16 (1) , 54. 10.3390/sym16010054

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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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Publisher: MDPI
ISSN: 2073-8994
Date of First Compliant Deposit: 1 January 2024
Date of Acceptance: 28 December 2023
Last Modified: 09 Jan 2024 12:15

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