A complete classification of three-dimensional algebras over ℝ and ℂ — 温故知新 (visiting old, learn new)

Abstract

We provide a complete classification of three-dimensional associative algebras over the real and complex number fields based on a completely elementary proof. We list up all the multiplication tables of the algebras up to isomorphism. We compare our results with those given by mathematicians in the 19th century and this century.