The class of algebras of Jordan type η was introduced by Hall, Rehren and Shpectorov in 2015 within the much broader class of axial algebras. Algebras of Jordan type are commutative algebras A over a field of characteristic not 2, generated by primitive idempotents, called axes, whose adjoint action on A has minimal polynomial dividing (x − 1)x(x − η) and where multiplication of eigenvectors follows the rules similar to the Peirce decomposition in Jordan algebras.Naturally, Jordan algebras generated by primitive idempotents are examples of algebras of Jordan type <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\eta={1\over 2}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>η</mi> <mo>=</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Further examples are given by the Matsuo algebras constructed from 3-transposition groups. These examples exist for all values of η ≠ 0, 1. Jordan algebras and (factors of) Matsuo algebras constitute all currently known examples of algebras of Jordan type and it is conjectured that there are no additional examples.In this paper we introduce the concept of a solid 2-generated subalgebra, as a subalgebra J such that all primitive idempotents from J are axes of A. We prove, for axes a,b ∈ A, that if the value of the Frobenius form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(a,b)\notin\{0,{1\over 4},1\}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∉</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </math></EquationSource> </InlineEquation> then J = 〈〈a, b〉〉 is solid, that is, generic 2-generated subalgebras are solid. Furthermore, in characteristic zero, J is solid even for the values (a, b) = 0, 1. As a corollary, in characteristic zero, either A has infinitely many axes and an infinite automorphism group, or it is a Matsuo algebra or a factor of Matsuo algebra.

Solid subalgebras in algebras of Jordan type half

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发表日期：

2026年3月26日

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Ilya Gorshkov,
Sergey Shpectorov,
Alexey Staroletov

摘要

The class of algebras of Jordan type η was introduced by Hall, Rehren and Shpectorov in 2015 within the much broader class of axial algebras. Algebras of Jordan type are commutative algebras A over a field of characteristic not 2, generated by primitive idempotents, called axes, whose adjoint action on A has minimal polynomial dividing (x − 1)x(x − η) and where multiplication of eigenvectors follows the rules similar to the Peirce decomposition in Jordan algebras.

Naturally, Jordan algebras generated by primitive idempotents are examples of algebras of Jordan type $\eta={1\over 2}$ $η = \frac{1}{2}$ . Further examples are given by the Matsuo algebras constructed from 3-transposition groups. These examples exist for all values of η ≠ 0, 1. Jordan algebras and (factors of) Matsuo algebras constitute all currently known examples of algebras of Jordan type and it is conjectured that there are no additional examples.

In this paper we introduce the concept of a solid 2-generated subalgebra, as a subalgebra J such that all primitive idempotents from J are axes of A. We prove, for axes a,b ∈ A, that if the value of the Frobenius form $(a,b)\notin\{0,{1\over 4},1\}$ $(a, b) \notin {0, \frac{1}{4}, 1}$ then J = 〈〈a, b〉〉 is solid, that is, generic 2-generated subalgebras are solid. Furthermore, in characteristic zero, J is solid even for the values (a, b) = 0, 1. As a corollary, in characteristic zero, either A has infinitely many axes and an infinite automorphism group, or it is a Matsuo algebra or a factor of Matsuo algebra.