相关论文: An expansion algorithm for constructing axial alge…
We extend the theory of Matsuo algebras, which are certain non-associative algebras related to 3-transposition groups, to characteristic 2. Instead of idempotent elements associated to points in the corresponding Fischer space, our algebras…
A class of axial algebras generated by two axes with eigenvalues 0, 1, $\eta$ and $\xi$ called axial algebras of Majorana type is introduced and classified when they are 2-generated, over fields of characteristics neither 2 nor 5 and there…
A code algebra $A_C$ is a non-associative commutative algebra defined via a binary linear code $C$. We study certain idempotents in code algebras, which we call small idempotents, that are determined by a single non-zero codeword. For a…
A Majorana algebra is a commutative nonassociative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some of the properties of the $2A$-axes of the Monster Griess algebra. The term was introduced by…
A code algebra $A_C$ is a nonassociative commutative algebra defined via a binary linear code $C$. In a previous paper, we classified when code algebras are $\mathbb{Z}_2$-graded axial (decomposition) algebras generated by small…
Axial algebras are a class of non-associative algebra with a strong link to finite (especially simple) groups which have recently received much attention. Of primary interest are the axial algebras of Monster type $(\alpha, \beta)$, of…
We consider Majorana algebras generated by three Majorana axes $a_0$, $a_1$ and $a_2$ such that $a_0$ and $a_1$ generate a dihedral algebra of type 2A. We show that such an algebra must occur as a Majorana representation of one of 27…
We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the Griess algebra whose automorphism group is…
``Fusion rules'' are laws of multiplication among eigenspaces of an idempotent. We establish fusion rules for flexible power-associative algebras, following Albert. We define the notion of an axis in the noncommutative setting (compare with…
Recently Takahiro Yabe gave an almost complete classification of primitive symmetric $2$-generated axial algebras of Monster type. In this note, we construct a new infinite-dimensional primitive $2$-generated symmetric axial algebra of…
Axial algebras are commutative nonassociative algebras generated by a finite set of primitive idempotents which action on an algebra is semisimple, and the fusion laws on the products between eigenvectors for these idempotents are…
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…
A pair $(G,T)$ is called a faithful odd transposition group if $T$ is a normal set of involutions generating the group $G$ and the product of any two distinct elements of $T$ has odd order. We introduce a special subclass of such groups, a…
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 $C$ be a binary linear code of length…
Ivanov introduced the shape of a Majorana algebra as a record of the $2$-generated subalgebras arising in that algebra. As a broad generalisation of this concept and to free it from the ambient algebra, we introduce the concept of an axet…
We provide the basic setup for the project, initiated by Felix Rehren, aiming at classifying all 2-generated axial algebras of Monster type $(\alpha,\beta)$ over a field $\mathbb F$. Using this, we first show that every such algebra has…
We study the class of idempotent-generated pseudo-composition algebras, which is a subclass of the family of axial algebras. More specifically, we utilise the group-algebra correspondence, natural to the axial framework in order to study…
Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type $(\alpha, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(\alpha,\frac{1}{2}, \delta)$…
We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The…
It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We…