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

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

Centroids and derivations of low-dimensional Leibniz algebra

In this paper we introduce the concept of centroid and derivation of Leibniz algebras. By using the classification results of Leibniz algebras obtained earlier, we describe the centroids and derivations of low-dimensional Leibniz algebras. We also study some properties of centroids of Leibniz algebras and use these properties to categorize the algebras to have so-called small centroids. The description of the derivations enables us to specify an important subclass of Leibniz algebras called characteristically nilpotent

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

(α, β, γ) - derivations of diassociative algebras

In this research we introduce a generalized derivations of diassociative algebras and study its properties. This generalization depends on some parameters. In this paper we specify all possible values of the parameters. We also provide all the generalized derivations of low-dimensional complex diassociative algebras

Algorithms for computations of Loday algebras’ invariants

The paper is devoted to applications of some computer programs to study structural determination of Loday algebras. We present how these computer programs can be applied in computations of various invariants of Loday algebras and provide several computer programs in Maple to verify Loday algebras’ identities, the isomorphisms between the algebras, as a special case, to describe the automorphism groups, centroids and derivations

Classification and derivations of low-dimensional complex dialgebras

The thesis is mainly comprised of two parts. In the first part we consider the classification problem of low-dimensional associative, diassociative and dendriform algebras. Since so far there are no research results dealing with representing diassociative and dendriform algebras in form of precise tables under some basis, it is desirable to have such lists up to isomorphisms. There is no standard approach to the classification problem of algebras. One of the approaches which can be applied is to fix a basis and represent the algebras in terms of structure constants. Due to the identities we have constraints for the structure constants in polynomial form. Solving the system of polynomials we get a redundant list of all the algebras from given class. Then we erase isomorphic copies from the list. It is slightly tedious to perform this procedure by hand. For this case we construct and use several computer programs. They are applied to verify the isomorphism between found algebras, to find automorphism groups and verify the algebra identities. In conclusion, we give complete lists of isomorphism classes for diassociative and dendriform algebras in low dimensions. We found for diassociative algebras four isomorphism classes (one parametric family and another three are single class) in dimension two, 17 isomorphism classes (one parametric family and others are single classes) in dimension three and for nilpotent diassociative algebras we obtain 16 isomorphism classes (all of them are parametric family) in dimension four. In dendriform algebras case there are twelve isomorphism classes (one parametric family and another eleven are single classes) in dimension two. The second part of the thesis is devoted to the computation of derivations of low-dimensional associative, diassociative and dendriform algebras. We give the derivations the above mentioned classes of algebras in dimensions two and three

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