On Ternary $F$-manifold Algebras and their Representations

TL;DR

This paper introduces ternary $F$-manifold algebras, explores their representations including duals, and constructs new algebraic structures using Rota-Baxter operators, expanding the theoretical framework of $F$-manifold algebras.

Contribution

It generalizes $F$-manifold algebras to ternary cases, develops their representation theory, and introduces methods to construct new algebras via Rota-Baxter operators.

Findings

Defined dual representations with additional conditions

Established coherence ternary $F$-manifold algebras

Constructed ternary pre-$F$-manifold algebras using Rota-Baxter operators

Abstract

We introduce a notion of ternary $F$-manifold algebras which is a generalization of $F$-manifold algebras. We study representation theory of ternary $F$-manifold algebras. In particular, we introduce a notion of dual representation which requires additional conditions similar to the binary case. We then establish a notion of a coherence ternary $F$-manifold algebra. Moreover, we investigate the construction of ternary $F$-manifold algebras using $F$-manifold algebras. Furthermore, we introduce and investigate a notion of a relative Rota-Baxter operator with respect to a representation and use it to construct ternary pre-$F$-manifold algebras.