Citation
Ahmed, Houida Mohammed Hussein
(2020)
Classification of twodimensional algebras using matrices of structure constants approach.
Doctoral thesis, Universiti Putra Malaysia.
Abstract
One of the main issues of modern algebra is the classification problem related to
finitedimensional 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 coordinatebased approach. The
abovementioned 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 ndimensional case cover only small
parts of all ndimensional 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 ndimensional
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 finitedimensional 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 twodimensional 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 twodimensional 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 twodimensional algebras are described and tabulated.
We also present all possible evolution algebra structures on twodimensional
vector space over any algebraically closed field. The description of group automorphisms
and derivation evolution algebras is given.
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