Polynomial invariants for low dimensional algebras

We classify all two-dimensional simple algebras (which may be non-associative) over an algebraically closed field. For each two-dimensional algebra $\mathcal{A}$, we describe a minimal (with respect to inclusion) generating set for the algebra of invariants of the $m$-tuples of $\mathcal{A}$ in the case of characteristic zero. In particular, we establish that for any two-dimensional simple algebra $\mathcal{A}$ with a non-trivial automorphism group, the Artin--Procesi--Iltyakov Equality holds for $\mathcal{A}^m$; that is, the algebra of polynomial invariants of $m$-tuples of $\mathcal{A}$ is generated by operator traces. As a consequence, we describe two-dimensional algebras that admit a symmetric or skew-symmetric invariant nondegenerate bilinear form.