The basic concepts of Majorana theory were introduced by A. A. Ivanov (2009) as a tool to examine the subalgebras of the Griess algebra V_M from an elementary axiomatic perspective. A Majorana algebra is a commutative non-associative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some properties of Conway's 2A-axes of V_M. If G is a finite group generated by a G-stable set of involutions T, a Majorana representation of (G,T) is an algebra representation of G on a Majorana algebra V together with a compatible bijection between T and a set of Majorana axes of V . Ivanov's definitions were inspired by Sakuma's theorem, which establishes that any two-generated Majorana algebra is isomorphic to one of the Norton-Sakuma algebras. Since then, the construction of Majorana representations of various finite groups has given non-trivial information about the structure of V_M. This thesis concerns two main themes within Majorana theory. The first one is related with the study of some low-dimensional Majorana algebras: the Norton-Sakuma algebras and the Majorana representations of the symmetric group of degree 4 of shapes (2A,3C) and (2B,3C). For each one of these algebras, all the idempotents, automorphism groups, and maximal associative subalgebras are described. The second theme is related with a Majorana representation V of the alternating group of degree 12 generated by 11,880 Majorana axes. In particular, the possible linear relations between the 3A-, 4A-, and 5A-axes of V and the Majorana axes of V are explored. Using the known subalgebras and the inner product structure of V , it is proved that neither sets of 3A-axes nor 4A-axes is contained in the linear span of the Majorana axes. When V is a subalgebra of V_M, these results, enhanced with information about the characters of the Monster group, establish that the dimension of V lies between 3,960 and 4,689.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:656601 |
| Date | January 2014 |
| Creators | Castillo Ramirez, Alonso |
| Contributors | Ivanov, Alexander |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/24696 |
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