Quantum deformations of associative algebras and integrable systems

Abstract

Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the WDVV equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton equations too. In particular, such deformations for the three-dimensional and infinite-dimensional algebras are described by the Boussinesq equation and KP hierarchy, respectively.

Numeration of the formulas is corrected

References in corpus (7)

• Relative formality theorem and quantisation of coisotropic submanifolds
• Geometry and Integrability of Topological-Antitopological Fusion
• Quantum cohomology of flag manifolds and Toda lattices
• Bihamiltonian Geometry, Darboux Coverings, and Linearization of the KP Hierarchy
• Coisotropic deformations of associative algebras and dispersionless integrable hierarchies
• Dispersionless integrable equations as coisotropic deformations. Extensions and reductions
• Theory of Submanifolds, Associativity Equations in 2D Topological Quantum Field Theories, and Frobenius Manifolds

Cited by in corpus (8)

• 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
• Classical $r$-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics
• Discrete integrable systems and deformations of associative algebras
• Discrete, q-difference deformations of associative algebras and integrable systems
• Continuous-discrete integrable equations and Darboux transformations as deformations of associative algebras
• Multidimensional the Ricci-flat spaces defined by nonlinear equations
• Menelaus relation and Fay's trisecant formula are associativity equations