Integral octonions: history and perspectives

Quadratic forms, Linear algebraic groups and Beyond. Wednesdays, 8:30-9:30am. 

https://uottawa-ca.zoom.us/j/99397061432?pwd=LzNBS1UrSVNJQ1ZiU28yWHAyNlV...

After setting the stage by recalling the basic properties of composition algebras over commutative rings, I sketch the history of intergral octonions, from its infancy in the 1860s to Coxeter’s groundbreaking paper of 1946. Inspired by results due to Mahler (1942) and Allcock (1999), I proceed to describe the one-sided ideal structure of octonion algebras over arbitrary commutative rings. The lecture concludes with a non-orthogonal version of the classical Cayley-Dickson construction that allows for a description of integral octonions (more precisely, of their multiplicative structure) in an intrinsic manner.