31 research outputs found

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

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    An axial algebra AA is a commutative non-associative algebra generated by primitive idempotents, called axes, whose adjoint action on AA 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 TT, every axis aa leads to a subgroup of automorphisms TaT_a of AA. The group generated by all TaT_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 22-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

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    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

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    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 CC be a binary linear code of length nn. A basis for the code algebra ACA_C consists of nn idempotents and a vector for each non-constant codeword of CC. 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 ss-map, which comes from the code structure. For a general code algebra, we classify the eigenvalues and eigenvectors of the smallest examples of the ss-map construction, and hence show that certain code algebras are axial algebras. We give some examples, including that for a Hamming code H8H_8 where the code algebra AH8A_{H_8} is an axial algebra and embeds in the code VOA VH8V_{H_8}.Comment: 32 pages, including an appendi

    Axial algebras of Jordan and Monster type

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    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

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    We study presentations, defined by Sidki, resulting in groups y(m,n)y(m,n) that are conjectured to be finite orthogonal groups of dimension m+1m+1 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 y(m)y(m) and explicitly relate this to previous work done by Sidki. The original groups y(m,n)y(m,n) can be found as quotients over congruence subgroups of y(m)y(m). We give two representations of our group y(m)y(m). One into an orthogonal group of dimension m+1m+1 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 SL2mβˆ’2(R)SL_{2^{m-2}}(R) is recovered and extended as an action on a submodule of the Clifford algebra

    Code algebras which are axial algebras and their Z2\mathbb{Z}_2-gradings

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    A code algebra ACA_C is a non-associative commutative algebra defined via a binary linear code CC. 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 CC, we show that small idempotents are primitive and semisimple and we calculate their fusion law. If CC is a projective code generated by a conjugacy class of codewords, we show that ACA_C is generated by small idempotents and so is, in fact, an axial algebra. Furthermore, we classify when the fusion law is Z2\mathbb{Z}_2-graded. In doing so, we exhibit an infinite family of Z2Γ—Z2\mathbb{Z}_2 \times \mathbb{Z}_2-graded axial algebras - these are the first known examples of axial algebras with a non-trivial grading other than a Z2\mathbb{Z}_2-grading.Comment: 29 page

    Automorphism groups of axial algebras

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    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
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