Classification of two-dimensional algebras using matrices of structure constants approach - Ahmed, Houida Mohammed Hussein (2020)
One of the main issues of modern algebra is the classification problem related to finite-dimensional algebras. So far, there are two approaches to deal with this problem. The first one is the structural approach (basis free and invariant). However, when dealing with general types of algebras, this approach shows that it is ineffective due to the difficulty in dealing with such general algebra types. The second approach that tried to deal with this particular problem is the coordinate-based approach. The above-mentioned approaches somehow are complementary to each other. Moreover, it is noticed that well known classification theorems of algebras, for example Lie, associative, Jordan and etc., in fixed n-dimensional case cover only small parts of all n-dimensional algebras. In fact, the part out of the consideration is much bigger and it is a dense in Zariski topology subset of the set of all n-dimensional algebras. Therefore, instead of the classification of some classes of algebras one can try to classify all algebras in fixed dimensions. We use a new approach to study the classification problem of finite dimensional algebras and the main advantage of the approach, proposed in Bekbaev (2015) which we apply it here, is the fact that it reduces the classification and some other problems of finite-dimensional algebras to the investigation of a system of equations which can be solved using some computation programs. We give a list of algebras, depending on their matrices of structure constants (MSC). Therefore, any two-dimensional algebra is considered isomorphic to one of the algebras in the provided list. The organization of the thesis starts with studying the problem of classification over what is known as algebraically closed fields of not two and not three characteristics. After that we presented the solution for this problem that deals with the state of classification to be over algebraically closed fields of characteristics two and three. In these cases, we provide a list of algebras, using their matrices of structure constants (MSC). Then the automorphism groups and derivation algebras for the all listed canonical algebras are described. We also present complete lists of isomorphism classes of all two-dimensional left (right) unital algebras, (not necessarily commutative) Jordan algebras, power associative algebras and we specify commutative Jordan, associative, Lie, Leibniz and Zinbiel algebras by providing the lists of canonical representatives of their structure constant’s matrices. All subalgebras, idempotents, ideals and the left quasiunits of two-dimensional algebras are described and tabulated. We also present all possible evolution algebra structures on two-dimensional vector space over any algebraically closed field. The description of group automorphisms and derivation evolution algebras is given.