Axial algebras of Monster type (2η, η)

Axial algebras are commutative nonassociative algebras generated by a set of special idempotents called axes whose adjoint maps are semisimple. The algebra product is controlled by a fusion law. Jordan algebras, the Griess algebra for the Monster group are some of the examples for axial algebras. This thesis is about study of axial algebras satisfying the Monster like fusion law where non unit elements α, β of the fusion law satisfy the condition α = 2β. This was an exceptional case in the literature. We show that axes satisfying the above fusion law arise as the sum of two orthogonal axes of Jordan type. These are called the axes of Monster type M(2η, η) or the double axes. Then we present the classification of 2-generated subalgebras of the Matsuo algebras generated by double axes. Further we construct two infinite classes of axial algebras for the symmetric group of S\(_{n}\) and for symplectic group Sp(2n, 2) which satisfy fusion law M(2η, η)