Two-dimensional Dendriform Algebra Structures over Any Base Field

In the paper we consider a class of dendriform algebras. A dendriform algebra is a generalization of associative algebra. It is typically defined as an algebra equipped by two multiplication operations satisfying certain compatibility conditions. These algebras have interesting representation theories, leading to insights in both algebra, geometry and physics. We provide a full classification, up to isomorphism, of all two-dimensinal dendriform algebras over any base field. Note that the solution to the classification problem of algebras essentially depends on properties of the base field. The results obtained so far were over the field of complex numbers only. These results come as a particular case of the results in this paper. Moreover, we revise the list of such representatives given earlier over the field of complex numbers.