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

TL;DR

This paper introduces a new scheme inspired by classical r-matrix formalism for constructing Frobenius manifolds and solutions to WDVV equations, utilizing the Rota-Baxter identity.

Contribution

It develops a general framework for creating flat pencils of metrics and Frobenius manifolds based on classical r-matrix techniques and illustrates it with algebra examples.

Findings

Constructs Frobenius manifolds from Laurent series and meromorphic functions.

Provides a scheme linking Rota-Baxter identity with Frobenius structures.

Offers explicit examples demonstrating the scheme's application.

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.