31 research outputs found
Whose Grass Is Greener? Green Marketing: Toward a Uniform Approach for Responsible Environmental Advertising
An axial algebra is a commutative non-associative algebra generated by
primitive idempotents, called axes, whose adjoint action on 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 , every axis leads to a subgroup of
automorphisms of . The group generated by all 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 -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
Code algebras, axial algebras and VOAs
Inspired by code vertex operator algebras (VOAs) and their representation
theory, we define code algebras, a new class of commutative non-associative
algebras constructed from binary linear codes. Let be a binary linear code
of length . A basis for the code algebra consists of idempotents
and a vector for each non-constant codeword of . We show that code algebras
are almost always simple and, under mild conditions on their structure
constants, admit an associating bilinear form. We determine the Peirce
decomposition and the fusion law for the idempotents in the basis, and we give
a construction to find additional idempotents, called the -map, which comes
from the code structure. For a general code algebra, we classify the
eigenvalues and eigenvectors of the smallest examples of the -map
construction, and hence show that certain code algebras are axial algebras. We
give some examples, including that for a Hamming code where the code
algebra is an axial algebra and embeds in the code VOA .Comment: 32 pages, including an appendi
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
On Sidki's presentation for orthogonal groups
We study presentations, defined by Sidki, resulting in groups that
are conjectured to be finite orthogonal groups of dimension in
characteristic two. This conjecture, if true, shows an interesting pattern,
possibly connected with Bott periodicity. It would also give new presentations
for a large family of finite orthogonal groups in characteristic two, with no
generator having the same order as the cyclic group of the field.
We generalise the presentation to an infinite version and explicitly
relate this to previous work done by Sidki. The original groups can be
found as quotients over congruence subgroups of . We give two
representations of our group . One into an orthogonal group of dimension
and the other, using Clifford algebras, into the corresponding pin group,
both defined over a ring in characteristic two. Hence, this gives two different
actions of the group. Sidki's homomorphism into is recovered
and extended as an action on a submodule of the Clifford algebra
Code algebras which are axial algebras and their -gradings
A code algebra is a non-associative commutative algebra defined via a
binary linear code . We study certain idempotents in code algebras, which we
call small idempotents, that are determined by a single non-zero codeword. For
a general code , we show that small idempotents are primitive and semisimple
and we calculate their fusion law. If is a projective code generated by a
conjugacy class of codewords, we show that is generated by small
idempotents and so is, in fact, an axial algebra. Furthermore, we classify when
the fusion law is -graded. In doing so, we exhibit an infinite
family of -graded axial algebras - these are
the first known examples of axial algebras with a non-trivial grading other
than a -grading.Comment: 29 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.Comment: 49 page