Superfun with Super-composition algebras and their application to Supersymmetric spacetimes and the Standard Model

The purpose of this talk would be to give a pedagogical introduction to composition algebras, superalgebras, and finally super-composition algebras. Hurwitz’s celebrated theorem states that there can be only seven composition algebras over the field of real numbers, namely the real numbers, the complex numbers, the quaternions, the octonions, and their split versions. Composition algebras play a crucial role in the classification of Lie and Jordan algebras, in number theory they are the backbone of many fundamental results like Lagrange’s identity (any number can be written as a sum of four squares), they also explain the geometry of Lorentzian spacetimes in 3, 4, 6, and 10 dimensions. In this talk while discussing these celebrated results, I will also try to introduce the super-version of the composition algebras, the associated Lie and Jordan superalgebras and how they might be related to the symmetries of the Standard Model and the geometry of supersymmetric spacetimes.