Polynomial invariants for low dimensional algebras

TL;DR

This work classifies all 2‑dimensional simple algebras over an algebraically closed field and analyzes the polynomial invariants of their m‑tuples under automorphisms. In characteristic zero it provides minimal generating sets for the invariant algebra $I_m(\mathcal{A})$ for each 2‑D algebra and shows that the API Equality holds for 2‑D simple algebras with nontrivial automorphism group, while giving explicit counterexamples when the automorphism group is trivial. The authors develop and apply standard invariant-theoretic techniques—reduction to multilinear invariants, polarization, and Noether bounds—to obtain concrete generators and trace-based invariants, including detailed trace formulas for the classified algebras. They further connect these invariants to bilinear forms, characterizing when symmetric or skew-symmetric invariant nondegenerate forms exist, and highlighting the role of automorphism groups in these properties. Overall, the paper extends deep invariant-theory methods to the landscape of 2‑dimensional algebras, yielding explicit generator sets and structural insights with potential applications to polynomial identities and forms on low-dimensional algebras.

Abstract

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.

Theorems & Definitions (55)

• Remark 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.5
• Lemma 2.7
• proof
• Lemma 3.1
• proof
• Proposition 3.2