2020
DOI: 10.48550/arxiv.2007.02430
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An infinite-dimensional 2-generated primitive axial algebra of Monster type

Abstract: Rehren proved in [13,14] that a primitive 2-generated axial algebra of Monster type (α, β), over a field of characteristic other than 2, has dimension at most eight if α / ∈ {2β, 4β}. In this note we construct an infinitedimensional 2-generated primitive axial algebra of Monster type (2, 1 2 ) over an arbitrary field F with char(F) = 2, 3. This shows that the second special case, α = 4β, is a true exception to Rehren's bound.

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Cited by 4 publications
(9 citation statements)
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“…Notice that z 2 and e are a 0-and α-eigenvector for z 1 , respectively. Since (z 1 , z 2 ) = 0 = (z 1 , e), we get (z 2 1 , z 2 ) = 0 = (z 1 , z 1 z 2 ) and (z 2 1 , e) = 0 = (z 1 , z 1 e). For the remaining two, we calculate:…”
Section: Let Us Summarisementioning
confidence: 94%
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“…Notice that z 2 and e are a 0-and α-eigenvector for z 1 , respectively. Since (z 1 , z 2 ) = 0 = (z 1 , e), we get (z 2 1 , z 2 ) = 0 = (z 1 , z 1 z 2 ) and (z 2 1 , e) = 0 = (z 1 , z 1 e). For the remaining two, we calculate:…”
Section: Let Us Summarisementioning
confidence: 94%
“…Since the form is symmetric, (z 2 1 , z 1 ) = (z 1 , z 2 1 ). Notice that z 2 and e are a 0-and α-eigenvector for z 1 , respectively.…”
Section: Let Us Summarisementioning
confidence: 99%
See 1 more Smart Citation
“…(n − 4). 5 Proof. We prove this by induction on n. One can easily check that this holds for the base cases n = 0, .…”
Section: Finite Dimensional Algebras On X(∞)mentioning
confidence: 99%
“…The highwater algebra H was introduced by Franchi, Mainardis and Shpectorov in [5] and also discovered independently by Yabe in [18]. It is an infinite dimensional 2-generated symmetric M(2, 1 2 )-axial algebra with axet X(∞) over any field of characteristic not 2, or 3.…”
Section: The Highwater Algebra and Its Characteristic 5 Covermentioning
confidence: 99%