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.

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