On the solutions to the Witten–Dijkgraaf–Verlinde–Verlinde associativity equations and their algebraic properties

Abstract

In this Letter I devise an algebraically feasible approach to investigating solutions to the oriented associativity equations, related with commutative and isoassociative algebras, interesting for applications in the quantum deformation theory and in some other fields of mathematics. The construction is based on a version of the Adler–Kostant–Symes scheme, applied to the Lie algebra of the loop diffeomorphism group of a torus and modified for the case of the Gauss–Manin displacement equations, depending on a spectral parameter. Their interpretation as characteristic equations for some system of the Lax–Sato type vector field equations made it possible to derive the determining separated Hamiltonian evolution equations for the related structure matrices, generating commutative and isoassociative algebras under regard.