Évènement

Abstract : Associativity is one of the most fundamental properties in mathematics. Almost all natural operations you can think of are associative : concatenation, union and intersection of sets, composition of functions etc. This property is so natural that it is often overlooked and we tend to forget that “(2+3)+4” and “2+(3+4)” don’t really have the same meaning.

But examples of non-associative algebras are quite easy to find. Using a process called the Cayley-Dickson Construction, you can enlarge the algebra of real numbers $\mathbb{R}$ to get the following inclusions of $\mathbb{R}$-algebras : $\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \mathbb{T} \subset …$, where all the algebras containing the algebra of the so-called octonions ($\mathbb{O}$) are no-longer associative (i.e. the multiplication is no longer associative).

In 1933, Pascual Jordan introduced a new class of algebras encompassing numerous examples of non-associative algebras (but also associative ones) in an attempt to formalize the notion of observables in quantum electrodynamics. Despite the fact that these algebras didn’t find many applications in physics, they became of great interest for mathematicians themselves.

In this talk I will try to give you a (hopefully) accessible introduction to the theory of non-associative algebras and Jordan algebras with the aim of stating the Classification Theorem for Finite-Dimensional Jordan Algebras.

PS : And if we have time, I will try to compute in front of you without my notes the very difficult calculations “(2+3)+4” and “2+(3+4)” (get ready because this is pretty impressive…).