bibkey: goncalves2018twodimensional
authors: Maria Inez Cardoso Goncalves; Daniel Goncalves; Dolores Martin Barquero; Candido Martin Gonzalez; Mercedes Siles Molina
year: 2018
title: "Squares and associative representations of two dimensional evolution algebras"
doi: 10.48550/arXiv.1807.02362
url: https://arxiv.org/abs/1807.02362v1
claim: "Lemma 3.15 and section 5.1 provide the algebra A_(8,alpha), equations for arbitrary natural bases, and its full automorphism group."
strata_touched:
  - D5/S3/QuadraticForms/EvolutionAlgebras/FiniteAutomorphismsInfiniteNaturalBases
license: citation-only
triage: anchor

The two-dimensional algebra and its older classification

The algebra $A_{8,1}$ has natural basis $u,v$, with $u^2=u$, $v^2=u$ and $uv=0$. Lemma 3.15 writes an arbitrary natural basis as $xu+yv$, $x^{\prime }u+y^{\prime }v$, with nonzero determinant and $x{x}^{\prime }+y{y}^{\prime }=0$. Section 5.1 begins from arbitrary invertible multiplication-preserving matrices and gives the full group ${\operatorname{diag}(1,1), \operatorname{diag}(1,-1)}$ in characteristic different from two. These are published ingredients, not new algebra or classification claims.

The cited passages do not assert infinitely many natural-basis classes modulo permutation and independent nonzero rescaling for this algebra. The Q14 application constructs and separates such an unbounded family. This distinction describes the bounded source reading, not a global literature census or a priority claim.

Related credited ingredients are arXiv:2303.00862v1, Theorem 26 and Remark 29 (the same full-group calculation), and arXiv:2006.14460v1, Corollary 2.13 and Proposition 2.16 (natural-basis block and extension structure). The infinite quotient in arXiv:2505.14584v1, Example 6.3, uses a degenerate algebra with an infinite full automorphism group and does not supply the Q14 conjunction.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.1807.02362
• URL: https://arxiv.org/abs/1807.02362v1
• Lemma 3.15: https://arxiv.org/html/1807.02362v1#S3.Thmlemma15
• Full group: https://arxiv.org/html/1807.02362v1#S5.SS1
• Table 3: https://arxiv.org/html/1807.02362v1#S5.T3

The source-admission reading establishes these locators and their scope. No external code is copied and no author-reported Lean check is used as current compiled evidence.