The spectra of finite 3-transposition groups

We calculate the spectrum of the diagram for each finite $3$-transposition group. Such graphs with a given minimal eigenvalue have occurred in the context of compact Griess subalgebras of vertex operator algebras

The hyperplanes of the U (4)(3) near hexagon

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U (4)(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative

Whose Grass Is Greener? Green Marketing: Toward a Uniform Approach for Responsible Environmental Advertising

An axial algebra $A$ is a commutative non-associative algebra generated by primitive idempotents, called axes, whose adjoint action on $A$ is semisimple and multiplication of eigenvectors is controlled by a certain fusion law. Different fusion laws define different classes of axial algebras. Axial algebras are inherently related to groups. Namely, when the fusion law is graded by an abelian group $T$, every axis $a$ leads to a subgroup of automorphisms $T_a$ of $A$. The group generated by all $T_a$ is called the Miyamoto group of the algebra. We describe a new algorithm for constructing axial algebras with a given Miyamoto group. A key feature of the algorithm is the expansion step, which allows us to overcome the $2$-closeness restriction of Seress's algorithm computing Majorana algebras. At the end we provide a list of examples for the Monster fusion law, computed using a MAGMA implementation of our algorithm.Comment: 31 page

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

Recovering the Lie algebra from its extremal geometry

An element $x$ of a Lie algebra $L$ over the field $F$ is extremal if $[x,[x,L]]=Fx$. Under minor assumptions, it is known that, for a simple Lie algebra $L$, the extremal geometry ${\cal{E}}(L)$ is a subspace of the projective geometry of $L$ and either has no lines or is the root shadow space of an irreducible spherical building $\Delta$. We prove that if $\Delta$ is of simply-laced type, then $L$ is a quotient of a Chevalley algebra of the same type.Comment: 24 page

On primitive axial algebras of Jordan type

In this note we give an overview of our knowledge regarding primitive axial algebras of Jordan type half and connections between $3$-transposition groups and Matsuo algebras. We also show that primitive axial algebras of Jordan type $\eta$ admit a Frobenius form, for any $\eta$.Comment: 10 page

Graphs 4n4_n that are isometrically embeddable in hypercubes

A connected 3-valent plane graph, whose faces are $q$- or 6-gons only, is called a {\em graph $q_n$}. We classify all graphs $4_n$, which are isometric subgraphs of a $m$-hypercube $H_m$.Comment: 18 pages, 25 drawing