Axial algebras of Monster type (2η,η) for D diagrams. I

Abstract

Axial algebras are a class of commutative algebras generated by idempotents with adjoint action semisimple and satisfying a prescribed fusion law. The class of Matsuo algebras was introduced by Matsuo and later generalized by Hall, Rehren, and Shpectorov. A Matsuo algebra M is built by a set of 3-transpositions D. Elements of D are idempotents in M and called axes. It is known that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type. In this paper, we study primitive subalgebras generated by a single axis and two double axes. We classify all such subalgebras in seven out of nine possible cases for a diagram on 3-transpositions that are involved in the generating elements. We also construct several infinite series of axial algebras of Monster type generalizing our 3-generated algebras.