The Classification Of Finite Dimensional Associative Algebras

The research on associative algebras aims to portray the structure and representations of all types of associative algebras.In the middle of the 19 th century,frobenius had proved that: there exists three types of finite dimensional division algebras over real numbers up to isomorphism,respectively are real numbers,complex numbers,quaternions.Few research on non-divisible algebras or algebras over algebraically closed field(such as complex numbers)had been made at home and abroad,with the dimension of algebras increasing,the complete classification of algebras has become difficult.In this article,we had study part of two types of problem mentioned above based on frobenius theorem and ring theory.First,we give the complete classification of 2-dimensional commutative associative algebras with identity over real number through the method of linear algebras and the representation of associative algebras,the result show that there are only three types of algebras up to isomorphism.Second,we give the complete classification of 3-dimensional associative algebras with principal generators over complex number up to isomorphism based on the equations satisfied by the principal generators and non-degenerate linear transformation,the result show that there are only three types of algebras up to isomorphism.Third,we give the complete classification of 4-dimensional associative algebras with principal generators over complex number up to isomorphism based on the similar methods as above.Four,we have discussed 5-dimensional associative algebras with principal generators over complex number,in order to study further,we have use translation,stretching to simplify the algebras.