Frobenius manifolds and Frobenius algebra-valued integrable systems

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper a new theory of Frobenius-algebra valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann, Kontsevich and Manin \cite{Kaufmann,KMK}. By specializing this construction, using a fixed Frobenius algebra $\mathcal{A},$ one can arrive at such a theory. More generally one can apply the same idea to construct an $\mathcal{A}$-valued Topological Quantum Field Theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed and, as an example, an $\mathcal{A}$-valued modified Camassa-Holm equation is constructed.