We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject.
Decomposition algebras and axial decomposition algebras are classes of commutative nonassociative algebras which are generalizations of axial algebras. The classes decomposition algebras, axial decomposition al;gebras and non-primitive axial algebras also have univesal algebras. Furthermore, by using the existence of a universal algebra, 2-generated non-primitive axial algebras of Jordan type is classified.
Axial algebras are non-associative algebras generated by semisimple idempotents whose adjoint actions obey a fusion law. Axial algebras that are generated by two such idempotents play a crucial role in the theory. We classify all primitive 2-generated axial algebras whose fusion laws have two eigenvalues and all graded primitive 2-generated axial algebras whose fusion laws have three eigenvalues. This represents a significant broadening in our understanding of axial algebras.
"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to primitive axial algebras, introduced recently by Hall, Rehren, and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and vertex operator algebras. In earlier work we studied primitive axial algebras, not necessarily commutative, and showed that they all have Jordan type. In this paper, we show that all finitely generated primitive axial algebras are direct sums of specifically described flexible finite dimensional noncommu…
A class of axial decomposition algebras with Miyamoto group generated by two Miyamoto automorphisms and three eigenvalues $0,1$ and $η$ is introduced and classified in the case with $η\notin\{0,1,\frac{1}{2}\}$. This class includes specializations of 2-generated axial algebras of Majorana type $(ξ,η)$ to the case with $ξ=η$.
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 the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3-transposition groups by relating modules over Matsuo algebras with representations of 3-transposition groups. As a by-product, we define, given a Fischer space, a group that can fulfill the role of a universal 3-transposit…
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.
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.
The notion of axial algebra is closely related to $3$-transposition groups, the Monster group and vertex operator algebras. In this work we continue our previous works and compete the proof that all algebras generated by a set of primitive axes not necessarily of the same type (see the definition in the body of the paper), are primitive axial algebras of Jordan 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 Mons…
Axial algebras are non-associative algebras generated by semisimple idempotents, known as axes, that all obey a fusion rule. Axial algebras were introduced by Hall, Rehren and Shpectorov as a generalisation of the axioms of Majorana theory, which was in turn introduced by Ivanov as an axiomatisation of certain properties of the 2A-axes of the Griess algebra. Axial algebras of Monster type are axial algebras whose axes obey the Monster, or Majorana, fusion rule. We construct an axial algebra of Monster type $M_{4A}$ over the polynomial ring $\mathbb{R}[t]$ that is generated by six axes whose Mi…
Axial algebras are a recently introduced class of non-associative algebra, with a naturally associated group, which generalise the Griess algebra and some key features of the moonshine VOA. Sakuma's Theorem classifies the eight $2$-generated axial algebras of Monster type. In this paper, we compute almost all the $3$-generated axial algebras whose associated Miyamoto group is minimal $3$-generated (this includes the minimal $3$-generated algebras). We note that this work was carried out independently to that of Mamontov, Staroletov and Whybrow and extends their result by computing more algebra…
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 …
An axial algebra over the field $\mathbb F$ is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of $\mathbb F$. Here we consider the first nonassociative case, where adjoint minimal polynomials divide $(x-1)x(x-η)$ for fixed $0\neqη\neq 1$. Jordan algebras arise when $η=\frac{1}{2}$, but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automor…
Axial algebras of Jordan type $η$ are commutative algebras generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-η)$, where $η\not\in\{0,1\}$ is fixed, with restrictive multiplication rules. These properties generalize the Pierce decompositions for idempotents in Jordan algebras, where $\frac{1}{2}$ is replaced with $η$. In particular, Jordan algebras generated by idempotents are axial algebras of Jordan type $\frac{1}{2}$. If $η\neq\frac{1}{2}$ then it is known that axial algebras of Jordan type $η$ are factors of the so-called Matsuo algebras corresp…
Nonassociative commutative algebras $A$ generated by idempotents $e$ whose adjoint operators ${\rm ad}_e\colon A \rightarrow A$, given by $x \mapsto xe$, are diagonalizable and have few eigenvalues are of recent interest. When certain fusion (multiplication) rules between the associated eigenspaces are imposed, the structure of these algebras remains rich yet rather rigid. For example vertex operator algebras give rise to such algebras. The connection between the Monster algebra and Monster group extends to many axial algebras which then have interesting groups of automorphisms. Axial algebras…
Axial algebras are a class of commutative algebras generated by idempotents, with adjoint action semisimple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren, and Shpectorov in 2015 as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster group. The class of Matsuo algebras was introduced by Matsuo and later generalized by Hall, Rehren, and Shpectorov. A Matsuo algebra $M$ is built by a set of 3-transpositions $D$. Elements of $D$ are idempotents in $M$ and called axes. In …
The Classification of the 2-generated Primitive Axial Algebras of Monster Type
Università Cattolica del Sacro Cuore（天主教撒罗库尔圣母大学）；Università degli Studi di Udine（乌迪内大学）；The Alan Turing Institute（艾伦·图灵研究所）；University of Bristol（布里斯托大学）
In this paper we prove that $2$-generated primitive axial algebras of Monster type $(2β, β)$ over a ring $R$ in which $2$ and $β$ are invertible can be generated as $R$-module by $8$ vectors. We then completely classify $2$-generated primitive axial algebras of Monster type $(2β, β)$ over any field of characteristic other than $2$.
A class of axial algebras generated by two axes with eigenvalues 0, 1, $η$ and $ξ$ 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 exists an automorphism switching generating axes.. The class includes deformations of nine Norton-Sakuma algebras. Over fields of characteristics 5, the axial algebras of Majorana type with the axial dimension at most 5 are clasified.
In this article, we develop a further adaptation of the method of Skjelbred-Sund to construct central extensions of axial algebras. We use our method to prove that all axial central extensions (with respect to a maximal set of axes) of complex simple finite-dimensional Jordan algebras are split and that all non-split axial central extensions of dimension $n\leq 4$ over an algebraically closed field of characteristic not $2$ are Jordan. Also, we give a classification of $2$-dimensional axial algebras and describe some important properties of these algebras.
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 general code $C$, we show that small idempotents are primitive and semisimple and we calculate their fusion law. If $C$ is a projective code generated by a conjugacy class of codewords, we show that $A_C$ is generated by small idempotents and so is, in fact, an axial algebra. Furthermore, we classify when the fusion law is $\mathbb{Z}_2$-graded. In doing so, we…
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 and shapes on an axet. A shape can be viewed as an algebra version of a group amalgam. Just like an amalgam, a shape leads to a unique algebra completion which may be non-trivial or it may collapse. Then for a natural family of shapes of generalised Monster type we classify all completion algebras and discover that a great majority of them collapse, confirmin…
Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov \cite{hrs,hrs1} as a broad generalisation of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional si…
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 Monster type $(2, \frac{1}{2})$ over a field of characteristic $5$, and use this algebra to complete the last case left open.
In the first half of this paper, we define axial algebras: nonassociative commutative algebras generated by axes, that is, semisimple idempotents---the prototypical example of which is Griess' algebra [C85] for the Monster group. When multiplication of eigenspaces of axes is controlled by fusion rules, the structure of the axial algebra is determined to a large degree. We give a construction of the universal Frobenius axial algebra on $n$ generators with a specified fusion rules, of which all $n$-generated Frobenius axial algebras with the same fusion rules are quotients. In the second half, w…
Mathematical Center in Akademgorodok（阿卡德戈罗德克数学中心）；Ministry of Science and Higher Education of the Russian Federation（俄罗斯联邦科学与高等教育部）；Sobolev Institute of Mathematics（索博列夫数学研究所）
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 $n$. A basis for the code algebra $A_C$ consists of $n$ idempotents and a vector for each non-constant codeword of $C$. 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 w…
Rehren proved that a primitive 2-generated axial algebra of Monster type $(α,β)$ has dimension at most eight if $α\notin\{2β,4β\}$. In this note we construct an infinite-dimensional 2-generated primitive axial algebra of Monster type $(2,\frac{1}{2})$ over an arbitrary field $F$ with $char(F)\neq 2,3$. This shows that the second special case, $α=4β$, is a true exception to Rehren's bound.
The Peirce decomposition of a Jordan algebra with respect to an idempotent is well known. This decomposition was taken one step further and generalized recently by Hall, Rehren and Shpectorov, withtheir introduction of {\it axial algebras}, and in particular {\it primitive axial algebras of Jordan type} (PJs for short). It turns out that these notions are closely related to $3$-transposition groups and vertex operator algebras. De Medts, Peacock, Shpectorov, and M. Van Couwenberghe generalized axial algebrasto {\it decomposition algebras} which, in particular, are not necessarily commutative. …
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 $(α, β)$, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. In this paper, we complete the classification of the symmetric $2$-generated primitive axial algebras of Monster type $(α, β)$. By previous work of Yabe, and Franchi and Mainardis, any such algebra is either explicitly known, or is a quotient of the infinite-dimensio…
"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and Vertex operator algebras. In this paper we consider fusion rules for semisimple idempotents, following Albert in the power-associative case. We examine the notion of an axis in the non-commutative setting and show that the dimension $d$ of any algebra $A$ generated by a pair $a,b$ of (not necessarily Jordan) axes o…
Axial algebras are commutative algebras generated by idempotents; they generalise associative algebras by allowing the idempotents to have additional eigenvectors, controlled by fusion rules. If the fusion rules are $\mathbb{Z}/2$-graded, axial algebras afford representations of transposition groups. We consider axial representations of Weyl groups of simply-laced root systems, which are examples of regular $3$-transposition groups. We introduce coset axes, a special class of idempotents based on embeddings of transposition groups, and use them to study the propagation of fusion rules in axial…
Let $R$ be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over $R$. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over $R$.
This paper presents some results on simple exceptional Jordan algebra over algebraically closed field $Φ$ with characteristic not 2. Namely an example of simple decomposition of $H(O_3)$ into the sum of two subalgebras of the type $H(Q_3)$ is produced, and it is shown that this decomposition is the only possible in terms of the type of simple subalgebras.
We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota--Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular we prove that there exists no post-Lie algebra structure on a pair $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is a simple Lie algebra and $\mathfrak{g}$ is a reductive Lie algebra, which is not isomorphic to $\mathfrak{n}$. We also show that there is no post-associative algebra structure on a pair $(A,B)$ arising from a …
The theory of direct decomposition of a centrally orthocomplete effect algebra into direct summands of various types utilizes the notion of a type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly) noncommutative version of an effect algebra. In this article we develop the basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set to PEAs, and show that TD sets induce decompositions of centrally orthocomplete PEAs into direct summands.
We classify all decompositions of $M_3(\mathbb{C})$ into a direct vector-space sum of two subalgebras such that one of the subalgebras contains the identity matrix.
We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from \cite{BezOl} and \cite{Kurochkin}. We also show that for an arbitrary infinite Steinitz number $s$ there exists a unital locally matrix algebra $A$ having the Steinitz number $s$ and not isomorphic to a tensor product of finite dimensional matrix algebras.
The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given.
Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf algebras, we determine necessary and sufficient conditions for this decomposition to be strongly graded.
