Enumerating 3-generated axial algebras of Monster type

An axial algebra is a commutative non-associative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion law as the Griess algebra. The $2$-generated axial algebras of Monster type, called Norton-Sakuma algebras, have been fully classified and are one of nine isomorphism types. In this paper, we enumerate and construct the $3$-generated axial algebras of Monster type which do not contain a $5\textrm{A}$, or $6\textrm{A}$ subalgebra.Comment: 27 pages. arXiv admin note: text overlap with arXiv:1804.0058

Computational Group Theory

This was the seventh workshop on Computational Group Theory. It showed that Computational Group Theory has significantly expanded its range of activities. For example, symbolic computations with groups and their representations and computations with infinite groups play a major role nowadays. The talks also presented connections and applications to cryptography, number theory and the algorithmic theory of algebras

Axial algebras of Jordan and Monster type

Axial algebras are a class of non-associative commutative algebras whose properties are defined in terms of a fusion law. When this fusion law is graded, the algebra has a naturally associated group of automorphisms and thus axial algebras are inherently related to group theory. Examples include most Jordan algebras and the Griess algebra for the Monster sporadic simple group. In this survey, we introduce axial algebras, discuss their structural properties and then concentrate on two specific classes: algebras of Jordan and Monster type, which are rich in examples related to simple groups.Comment: 39 page

Automorphism groups of axial algebras

Axial algebras are a class of commutative non-associative algebras which have a natural group of automorphisms, called the Miyamoto group. The motivating example is the Griess algebra which has the Monster sporadic simple group as its Miyamoto group. Previously, using an expansion algorithm, about 200 examples of axial algebras in the same class as the Griess algebra have been constructed in dimensions up to about 300. In this list, we see many reoccurring dimensions which suggests that there may be some unexpected isomorphisms. Such isomorphisms can be found when the full automorphism groups of the algebras are known. Hence, in this paper, we develop methods for computing the full automorphism groups of axial algebras and apply them to a number of examples of dimensions up to 151

Automorphism groups of axial algebras

Axial algebras are a class of commutative non-associative algebras which have a natural group of automorphisms, called the Miyamoto group. The motivating example is the Griess algebra which has the Monster sporadic simple group as its Miyamoto group. Previously, using an expansion algorithm, about 200 examples of axial algebras in the same class as the Griess algebra have been constructed in dimensions up to about 300. In this list, we see many reoccurring dimensions which suggests that there may be some unexpected isomorphisms. Such isomorphisms can be found when the full automorphism groups of the algebras are known. Hence, in this paper, we develop methods for computing the full automorphism groups of axial algebras and apply them to a number of examples of dimensions up to 151.Comment: 49 page

Majorana algebras and subgroups of the Monster

Majorana theory was introduced by A. A. Ivanov as an axiomatisation of certain properties of the 2A axes of the Griess algebra. This work was inspired by that of S. Sakuma who reproved certain important properties of the Monster simple group and the Griess algebra using the framework of vertex operator algebras. The objects at the centre of Majorana theory are known as Majorana algebras and are real, commutative, non-associative algebras that are generated by idempotents known as Majorana axes. To each Majorana axis, we associate a unique involution in the automorphism group of the algebra, known as a Majorana involution. These involutions form an important link between Majorana theory and group theory. In particular, Majorana algebras can be studied either in their own right or as Majorana representations of finite groups. The main aim of this work is to classify and construct Majorana algebras generated by three axes such that the subalgebra generated by two of these axes is isomorphic to a 2A dihedral subalgebra of the Griess algebra. We first show that such an algebra must occur as a Majorana representation of one of 26 subgroups of the Monster. These groups coincide with the list of triangle-point subgroups of the Monster given by S. P. Norton. In particular, our result reproves the completeness of Norton's list. This work builds on that of S. Decelle. Next, inspired by work of A. Seress, we design and implement an algorithm to construct the Majorana representations of a given group. We use this to construct a number of important Majorana representations which are independent of the main aim of this work. Finally, we use this algorithm along with our first result to construct all possible Majorana algebras generated by three axes, two of which generate a 2A-dihedral algebra. We use these constructions to show that each of these algebras must be isomorphic to a subalgebra of the Griess algebra. This is our main result and can equivalently be thought of as the construction of the subalgebras of the Griess algebra which correspond to the groups in Norton's list of triangle-point groups.Open Acces