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Towards the classification of unital 7-dimensional commutative algebras

An algebra is a vector space with a compatible product operation. An algebra is called commutative if the product of any two elements is independent of the order in which they are multiplied. A basic problem is to determine how many unital commutative algebras exist in a given dimension and to find all of these algebras. This classification problem has its origin in number theory and algebraic geometry. For dimension less than or equal to 6, Poonen has completely classified all unital commutative algebras up to isomorphism. For dimension greater than or equal to 7, the situation is much more complicated due to the fact that there are infinitely many algebras up to isomorphism. The purpose of this work is to develop new techniques to classify unital 7-dimensional commutative algebras up to isomorphism. An algebra is called local if there exists a unique maximal ideal m. Local algebras are basic building blocks for general algebras as any finite dimensional unital commutative algebra is isomorphic to a direct sum of finite dimensional unital commutative local algebras. Hence, in order to classify all finite dimensional unital commutative algebras, it suffices to classify all finite dimensional unital commutative local algebras. In this article, we classify all unital 7-dimensional commutative local algebras up to isomorphism with the exception of the special case k1 = 3 and k2 = 3, where, for each positive integer i, m i is the subalgebra generated by products of i elements in the maximal ideal m and ki is the dimension of the quotient algebram/m. When k2 = 1, we classify all finite dimensional unital commutative local algebras up to isomorphism. As a byproduct of our classification theorems, we discover several new classes of unital finite dimensional commutative