An infinite-dimensional 2-generated primitive axial algebra of Monster type

Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type (,) over a field of characteristic other than 2, has dimension at most 8 if ∉{2,4} if α∉{2β,4β}. In this note, we show that Rehren’s bound does not hold in the case α=4β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type (2,1/2) over an arbitrary field of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.