The description of dendriform algebra structures on two-dimensional complex space

In this paper, we classify all dendriform algebra structures on two-dimensional complex space. We distinguish twelve isomorphism classes (one parametric family and eleven concrete) of two-dimensional complex dendriform algebras, and show that they exhaust all possible cases

Solvable Leibniz algebras with NFn⊕ F1m nilradical

All finite-dimensional solvable Leibniz algebras L, having N = NFn⊕ F1m as the nilradical and the dimension of L equal to n+m+3 (the maximal dimension) are described. NFn and F1m are the null-filiform and naturally graded filiform Leibniz algebras of dimensions n and m, respectively. Moreover, we show that these algebras are rigid

Diassociative algebras and their derivations

The paper concerns the derivations of diassociative algebras. We introduce one important class of diassociative algebras, give simple properties of the right and left multiplication operators in diassociative algebras. Then we describe the derivations of complex diassociative algebras in dimension two and three

On derivations of low dimensional associative dialgebras

This study introduces a new algorithm for finding derivation of associative dialgebras. The algorithm applied to compute the derivations of low dimensional associative dialgebras. Some simple properties of derivation in terms of left multiplication operators also are provided

Four-dimensional nilpotent diassociative algebras

The paper is devoted to structural properties of diassociative algebras. We introduce the notions of nilpotency, solvability of the diassociative algebras and study their properties. The list of all possible nilpotent diassociative algebra structures on four-dimensional complex vector spaces is given

Classification of three dimensional complex Leibniz algebras

The aim of this paper is to complete the classification of three-dimensional complex Leibniz algebras. The description of isomorphism classes of three-dimensional complex Leibniz algebras has been given by Ayupov and Omirov in 1999. However, we found that this list has a little redundancy. In this paper we apply a method which is more elegant and it gives the precise list of isomorphism classes of these algebras.We compare our list with that of Ayupov-Omirov and show the corrections which should be made

Classification of 3-dimensional complex diassociative algebras

The paper deals with the classification problems of a subclass of finite-dimensional algebras. One considers a class of algebras having two algebraic operations with five identities. They have been called diassociative algebras by Loday. In this paper we describe all diassociative algebra structure in complex vector space of dimension at most three