# Fredrik Cumlin: Geometric interpretation of non-associative composition algebras

BSc thesis presentation

**Time: **
Mon 2020-06-01 10.30 - 11.30

**Location: **
Zoom, meeting ID: 647 0784 5619

**Participating: **
Fredrik Cumlin

**Supervisor: **
Wushi Goldring

### Abstract

This paper aims to discuss the connection between non-associative composition algebra and geometry. It will first recall the notion of an algebra, and investigate the properties of an algebra together with a composition norm. The composition norm will induce a law on the algebra, which is stated as the composition law. This law is then used to derive the multiplication and conjugation laws, where the last is also known as convolution. These laws are then used to prove Hurwitz’s celebrated theorem concerning the different finite composition algebras.

More properties of composition algebras will be covered, in order to look at the structure of the quaternions H and octonions O. The famous Fano plane will be the finishing touch of the relationship between the standard orthogonal vectors which construct the octonions.

Lastly, the notion of invertible maps in relation to invertible loops will be covered, to later show the connection between 8−dimensional rotations and multiplication of unit octonions.