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Polynomial invariants for 3-dimensional Leibniz algebras

Ivan Kaygorodov, Artem Lopatin

TL;DR

The paper classifies polynomial invariants and automorphism groups for all 3-dimensional non-Lie Leibniz algebras over $\mathbb{C}$, completing and aligning the ${\bf L}_3$ classification with prior work. It derives explicit generating sets for the invariant algebras $I_m(\mathcal{L})$ for each $\mathcal{L}\in{\bf L}_3$ and analyzes when the Artin–Procesi–Iltyakov equality holds. A key finding is that any two non-nilpotent algebras in ${\bf L}_3$ can be distinguished by traces of degrees at most $2$ and by the dimension of their automorphism groups, providing practical, low-degree invariants to distinguish isomorphism classes. The results advance invariant theory for Leibniz algebras by furnishing concrete, computable invariants tied to the algebra’s multiplication and automorphism structure.

Abstract

For each 3-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent 3-dimensional non-Lie Leibniz algebras can be distinguished by the traces of degrees $\leqslant 2$ and by the dimensions of their automorphism groups.

Polynomial invariants for 3-dimensional Leibniz algebras

TL;DR

The paper classifies polynomial invariants and automorphism groups for all 3-dimensional non-Lie Leibniz algebras over , completing and aligning the classification with prior work. It derives explicit generating sets for the invariant algebras for each and analyzes when the Artin–Procesi–Iltyakov equality holds. A key finding is that any two non-nilpotent algebras in can be distinguished by traces of degrees at most and by the dimension of their automorphism groups, providing practical, low-degree invariants to distinguish isomorphism classes. The results advance invariant theory for Leibniz algebras by furnishing concrete, computable invariants tied to the algebra’s multiplication and automorphism structure.

Abstract

For each 3-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent 3-dimensional non-Lie Leibniz algebras can be distinguished by the traces of degrees and by the dimensions of their automorphism groups.
Paper Structure (14 sections, 15 theorems, 40 equations)

This paper contains 14 sections, 15 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Algebra of invariants
  3. Leibniz algebras
  4. Results
  5. Notations
  6. Preliminaries on invariants
  7. Generic elements
  8. Operator traces
  9. Generators for algebras of invariants
  10. Traces
  11. Reduction to multilinear case
  12. Automorphisms of $3$-dimensional Leibniz algebras
  13. Generators for algebras of invariants
  14. Trace invariants

Key Result

Proposition 4

Assume that $m\geqslant1$ and a homogeneous monomial $h\in \mathbb{F}\langle \chi_0,\ldots,\chi_m\rangle$ of degree 1 in $\chi_{0}$ satisfies the equality $h= P^1_{\chi_{r_1}} \circ \cdots \circ P^k_{\chi_{r_k}} (\chi_0)$ for some symbols $P^1,\ldots,P^k\in\{L,R\}$, $1\leqslant r_1,\ldots,r_k\leqsla

Theorems & Definitions (27)

  • Definition 1
  • Remark 2
  • Definition 3
  • Proposition 4: Proposition 3.2 of Alvarez_Lopatin_2025
  • Proposition 5: Proposition 3.3 of Alvarez_Lopatin_2025
  • Proposition 6: cf. II.5, Theorem 2.5A of Weyl_book
  • Lemma 7: cf. Theorem 5.14 of Bruns_Gubeladze_book_2009
  • Lemma 8: see Theorem 7.6 of Alvarez_Lopatin_2025
  • Theorem 9
  • proof
  • ...and 17 more