Classification of $3$-dimensional unital commutative algebras

Abstract

We present the complete classification of non-isomorphic complex $3$-dimensional unital commutative algebras. Every such algebra decomposes canonically as $\mathbb{C}{e}_1\oplus V$, where $e_1$ is the unit and $V$ is the kernel of the trace form; the induced traceless $2$-dimensional commutative algebra on $V$ is one of four algebras, and the classification reduces to the classification of symmetric bilinear forms on $V$ up to the automorphisms of this kernel. The answer consists of six algebras, three one-parameter families, one two-parameter family and one three-parameter family.