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 talk based on a joint work with Ian strachan, 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 (Int Math
Res Not 19:929–952, 1996), Kontsevich and Manin (Inv Math 124: 313–339, 1996).
By specializing this construction, using a fixed Frobenius algebra A, one can arrive at
such a theory. More generally, one can apply the same idea to construct an 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 A-valued
modified Camassa–Holm equation is constructed.