Algebra Over A Field - Classification of Low-dimensional Algebras

Classification of Low-dimensional Algebras

Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by Eduard Study.

There exist two two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and a. According to the definition of an identity element,

It remains to specify

for the first algebra,

for the second algebra.

There exist five three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), a and b. Taking into account the definition of an identity element, it is sufficient to specify

for the first algebra,

for the second algebra,

for the third algebra,

for the fourth algebra,

for the fifth algebra.

The fourth algebra is non-commutative, others are commutative.