The Standard Model as an extension of the noncommutative algebra of forms

The Standard Model of particle physics can be deduced from a small number of axioms within Connes' noncommutative geometry (NCG). Boyle and Farnsworth [New J. Phys. 16 (2014) 123027] proposed to interpret Connes' approach as an algebra extension in the sense of Eilenberg. By doing so, they could deduce three axioms of the NCG Standard Model (i.e. order zero, order one and massless photon) from the single requirement that the extended algebra be associative. However, their approach was only applied to the finite part of the model because it fails for the full model. By taking into account the differential graded structure of the algebra of noncommutative differential forms, we obtain a formulation where the same three axioms are deduced from the associativity of the extended differential graded algebra, but which is now compatible with the full Standard Model.