The properties on F-manifold color algebras and pre-F-manifold color algebras

The concept of F-manifold algebras is an algebraic expression of F-manifolds. In this paper, we provide the definition of an F-manifold color algebra, which can be viewed as a natural generalization of an F-manifold algebra. We develop the representation theory of an F-manifold color algebra and show that F-manifold color algebras admitting non-degenerate symmetric bilinear forms are coherence F-manifold color algebras. The concept of pre-F-manifold color algebras is also presented, and using this definition one can construct F-manifold color algebras. These results extend some properties of F-manifold algebras.