On the structure of axial algebras

Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and simplicity; and (2) sum decompositions.Comment: 27 page

Probabilistic characterizations of some finite ring of matrices and its zero divisor graph

Let R be a finite ring. In this study, the probability that two random elements chosen from a finite ring have product zero is determined for some finite ring of matrices over Zn. Then, the results are used to construct the zero divisor graph which is defined as a graph whose vertices are the nonzero zero divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0

M-axial algebras related to 4-transposition groups

The main result of this thesis concerns the classification of 3-generated M-axial algebras A such that every 2-generated subalgebra of A is a Sakuma algebra of type NX, where N∈{2, 3, 4} and X∈{A, B, C}. This goal requires the classification of all groups $G$ which are quotients of the groups T$^($$^s$$^1$$^,$ $^s$$^2$$^,$ $^s$$^3$$^)$ = for s$_1$, s$_2$, s$_3$ ∈{3, 4} and the set of all conjugates of x, y and z satisfies the 4-transposition condition. We show that those groups are quotients of eight groups. We show which of these eight groups can be generated by Miyamoto involutions. This can be done by classifying all possible M-axial algebras for them. In addition, we discuss the embedding of Fisher spaces into a vector space over GF(2) in Chapter 3

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 a subclass of 3-generated axial algebras of Monster type in terms of their groups and shapes. It turns out that the vast majority of the possible shapes for such algebras collapse; that is they do not lead to non-trivial examples. This is in sharp contrast to previous thinking. Accordingly, we develop a method of minimal forbidden configurations, to allow us to efficiently recognise and eliminate collapsing shapes

On the probability and graph of some finite rings of matrices

The study on probability theory in finite rings has been an interest of various researchers. One of the probabilities that has caught their attention is the probability that two elements of a ring have product zero. In this study, the probability is determined for a finite ring R of matrices over integers modulo four. First, the annihilators of R are determined with the assistance of Groups, Algorithms and Programming (GAP) software and then the probability is calculated using the definition. Next, by using the results obtained, the zero divisor graph of the ring R is constructed. A zero divisor graph is defined as a graph in which the zero divisors of R are its vertices and two vertices are connected by an edge if their product is zero