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Level: BSc, MSci, MSc
Title: Octonions and other non-associative algebras
Supervisor:

Prof. Shahn Majid
Research Area: Geometry and Analysis
Description:

The octonions O are commonly agreed as what come next in the family R, C, H, O where R means the usual real numbers, C the complex numbers and H the skew-field of quaternions. The quaternions are non-commutative -- it matters in what order you multiply them, while the octonions go one step further and are nonassociative -- it matters in which way you bracket their products [1]. In 1999 a new way of thinking about the octonions was introduced [2] in which the nonassociativity is strictly controlled by a group cocycle and which can be seen as saying that in fact the octonions are associative but in a modified sense of a certain monoidal category. The project will recount this new approach and explain this construction. The MSci version would be expected to consider generalisations to what comes next after O, such as the 16-onions given by the Cayley-Dixon process and/or enlarge on the categorical picture.

  1. J. H. Conway, D. Smith, On Quaternions and Octonions. CRC Press, USA, 2003.
  2. H. Albuquerque, S. Majid, Quasialgebra Structure of the Octonions, J. Algebra 220 (1999) 188–224.
Further Reading:
Key Modules:

To be confirmed with the supervisor.
Other Information:
Current Availability: Yes