# Axial algebras

Rehren, Felix Gabriel (2015). Axial algebras. University of Birmingham. Ph.D.

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## Abstract

Axial algebras are nonassociative algebras controlled by fusion rules for idempotents. We have three main results. Firstly, we give a classification of axial algebras with fusion rules of Jordan type, with parameter alpha, in terms of 3-transposition groups. When alpha is 1/2, we also classify the related Jordan algebras. Secondly, we develop a structure theory for Matsuo algebras, especially using large associative subalgebras, and apply it to the special case of the Dynkin diagram of type A$$_n$$, which has relations to vertex operator algebras. Thirdly, we generalize dihedral axial algebras of Ising type, with parameters alpha and beta, coming from the Monster sporadic simple group. This also helps determine the role that the parameters play in the larger theory, where indeed the Griess algebra turns out to be a special point.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Shpectorov, SergeyUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: None/not applicable
Subjects: Q Science > QA Mathematics
URI: http://etheses.bham.ac.uk/id/eprint/5948

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