Classical $r$-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics

Abstract

A general scheme for construction of flat pencils of contravariant metrics and Frobenius manifolds as well as related solutions to WDVV associativity equations is formulated. The advantage is taken from the Rota-Baxter identity and some relation being counterpart of the modified Yang-Baxter identity from the classical $r$-matrix formalism. The scheme for the construction of Frobenius manifolds is illustrated on the algebras of formal Laurent series and meromorphic functions on Riemann sphere.

article, 46 pages, v3: small revision, v2: preliminary settings in sections 4 and 5 are clarified and corrected, the rest is accordingly modified

References in corpus (5)

• The Extended Bigraded Toda hierarchy
• Coisotropic deformations of associative algebras and dispersionless integrable hierarchies
• Integrable hierarchies and the mirror model of local CP1
• Quantum deformations of associative algebras and integrable systems
• On classification and construction of algebraic Frobenius manifolds

Cited by in corpus (4)

• Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities
• An extension of the Kadomtsev-Petviashvili hierarchy and its Hamiltonian structures
• Infinite-dimensional Frobenius Manifolds Underlying the Universal Whitham Hierarchy
• Infinite-dimensional Frobenius Manifolds Underlying the Toda Lattice Hierarchy