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

Abstract

AbstractRehren 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 $$(\alpha ,\beta )$$ ( α , β ) , over a field of characteristic other than 2, has dimension at most 8 if $$\alpha \notin \{2\beta ,4\beta \}$$ α ∉ { 2 β , 4 β } . In this note, we show that Rehren’s bound does not hold in the case $$\alpha =4\beta $$ α = 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type $$(2,\frac{1}{2})$$ ( 2 , 1 2 ) over an arbitrary field $${{\mathbb {F}}}$$ F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.