Griess Algebra Research Articles

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.

Quotients of the Highwater algebra and its cover

Primitive axial algebras of Monster type are a class of non-associative algebras with a strong link to finite (especially simple) groups. The motivating example is the Griess algebra, with the Monster as its automorphism group. A crucial step towards the understanding of such algebras is the explicit description of the 2-generated symmetric objects. Recent work of Yabe, and Franchi and Mainardis shows that any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra H, or its characteristic 5 cover Hˆ.In this paper, we complete the classification of symmetric axial algebras of Monster type by determining the quotients of H and Hˆ. We proceed in a unified way, by defining a cover of H in all characteristics. This cover has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic 5.

2-generated axial algebras of Monster type (2β,β)

Axial algebras of Monster type (α,β) are a class of non-associative algebras that includes, besides associative algebras, other important examples such as the Jordan algebras and the Griess algebra. 2-generated primitive axial algebras of Monster type (α,β) naturally split into three cases: the case when α∉{2β,4β}, the case α=4β and α=2β. In this paper we give a complete classification all 2-generated primitive axial algebras of Monster type (2β,β).

Monstrous M-Theory

In 26+1 space–time dimensions, we introduce a gravity theory whose massless spectrum can be acted upon by the Monster group when reduced to 25+1 dimensions. This theory generalizes M-theory in many respects, and we name it Monstrous M-theory, or M2-theory. Upon Kaluza–Klein reduction to 25+1 dimensions, the M2-theory spectrum irreducibly splits as 1 ⊕ 196,883, where 1 is identified with the dilaton, and 196,883 is the dimension of the smallest non-trivial representation of the Monster. This provides a field theory explanation of the lowest instance of the Monstrous Moonshine, and it clarifies the definition of the Monster as the automorphism group of the Griess algebra by showing that such an algebra is not merely a sum of unrelated spaces, but descends from massless states for M2-theory, which includes Horowitz and Susskind’s bosonic M-theory as a subsector. Further evidence is provided by the decomposition of the coefficients of the partition function of Witten’s extremal Monster SCFT in terms of representations of SO24, the massless little group in 25+1; the purely bosonic nature of the involved SO24-representations may be traced back to the unique feature of 24 dimensions, which allow for a non-trivial generalization of the triality holding in 8 dimensions. Last but not least, a certain subsector of M2-theory, when coupled to a Rarita–Schwinger massless field in 26+1, exhibits the same number of bosonic and fermionic degrees of freedom; we cannot help but conjecture the existence of a would-be N=1 supergravity theory in 26+1 space–time dimensions.

Enumerating 3-generated axial algebras of Monster 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 Monster type in terms of their groups and shapes. It turns out that the vast majority of the possible shapes for such algebras collapse; that is they do not lead to non-trivial examples. This is in sharp contrast to previous thinking. Accordingly, we develop a method of minimal forbidden configurations, to allow us to efficiently recognise and eliminate collapsing shapes.

Double axes and subalgebras of Monster type in Matsuo algebras

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 (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 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 simple groups, and Matsuo algebras corresponding to 3-transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type Primitive subalgebras generated by two single or double axes are completely classified and 3-generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras.

Decomposition algebras and axial algebras

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.

On vertex operator algebras associated to Jordan algebras of Hermitian type

In this paper, we construct a family of vertex operator algebras parameterized by a non-zero complex number r, whose Griess algebras V2 are isomorphic to Hermitian Jordan algebras of type C and A with . This extends the construction of Ashihara and Miyamoto for the type B case, therefore we can realize all Hermitian Jordan algebras as the Griess algebras of moonshine type vertex operator algebras. Our construction is analogous to the construction of the vertex operator algebra given by Kac and Radul. For all values of r we also prove the corresponding simplicity results of .

3-dimensional Griess Algebras and Miyamoto Involutions

We consider a series of VOAs generated by 3-dimensional Griess algebras. We will show that these VOAs can be characterized by their 3-dimensional Griess algebras and their structures are uniquely determined. As an application, we will determine the groups generated by the Miyamoto involutions associated to Virasoro vectors of our VOAs.

Minimal 3-generated Majorana algebras

Majorana theory was introduced by A.A. Ivanov as the axiomatization of certain properties of the 2A-axes of the Griess algebra. Since its inception, Majorana theory has proved to be a remarkable tool with which to study objects related to the Griess algebra and the Monster simple group. We introduce the definition of a minimal 3-generated Majorana algebra and begin the first steps towards classifying such algebras.In particular, we give a complete classification of finite minimal 3-generated 6-transposition groups. We then use an algorithm developed in GAP by M. Pfeiffer and M. Whybrow, together with some additional computational tools, to give an almost complete description of all minimal 3-generated Majorana algebras arising from this list of groups.

Modules over Axial Algebras

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-transposition group.

Vertex operator algebras generated by Ising vectors of $$\sigma $$ σ -type

We prove the uniqueness of the simple vertex operator algebra of OZ-type generated by Ising vectors of $$\sigma $$ -type. We also prove that the simplicity can be omitted if the Griess algebra is isomorphic to the Matsuo algebra associated with the root system of type $$A_n$$ .

A correction of the decomposability result in a paper by Meyer–Neutsch

In their paper of 1993, Meyer and Neutsch established the existence of a 48-dimensional associative subalgebra in the Griess algebra G. By exhibiting an explicit counter example, the present paper shows a gap in the proof one of the key results in Meyer and Neutsch's paper, which states that an idempotent a in the Griess algebra is indecomposable if and only its Peirce 1-eigenspace (i.e. the 1-eigenspace of the linear transformation La:x↦ax) is one-dimensional. The present paper fixes this gap, and shows a more general result: let V be a real commutative nonassociative algebra with an associative inner product, and let c be a nonzero idempotent of V such that its Peirce 1-eigenspace is a subalgebra; then, c is indecomposable if and only if its Peirce 1-eigenspace is one-dimensional. The proof of this result is based on a general variational argument for real commutative metrised algebras with inner product.

Simplicities of VOAs associated to Jordan algebras of type B and character formulas for simple quotients

In this paper we study the VOA VJ,r constructed by Ashihara and Miyamoto in [2], which satisfies (VJ,r)0=C1,(VJ,r)1={0}, and the Griess algebra (VJ,r)2 is isomorphic to the type B simple Jordan algebra J. We construct simple quotients V¯J,r for r∈Z≠0 using dual-pair type constructions, and we reprove that VJ,r is simple if r∉Z. We also compute the character formula of the simple quotients V¯J,r for r=−2n,n∈Z>0.

Generalised dihedral subalgebras from the Monster

The conjugacy classes of the Monster which occur in the McKay observation correspond to the isomorphism types of certain 2 2 -generated subalgebras of the Griess algebra. Sakuma, Ivanov and others showed that these subalgebras match the classification of vertex algebras generated by two Ising conformal vectors, or of Majorana algebras generated by two axes. In both cases, the eigenvalues α , β \alpha ,\beta parametrising the theory are fixed to 1 4 \frac {1}{4} , 1 32 \frac {1}{32} . We generalise these parameters and the algebras which depend on them, in particular finding the largest possible (nonassociative) axial algebras which satisfy the same key features, by working extensively with the underlying rings. The resulting algebras admit an associating symmetric bilinear form and satisfy the same 6 6 -transposition property as the Monster; 1 4 \frac {1}{4} , 1 32 \frac {1}{32} turns out to be distinguished.

Primitive axial algebras of Jordan type

An axial algebra over the field 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 F. Here we consider the first nonassociative case, where adjoint minimal polynomials divide (x−1)x(x−η) for fixed 0≠η≠1. Jordan algebras arise when η=12, 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 automorphisms like those defined by Miyamoto exist, and there classify the 2-generated examples. Always for η≠12 and in identifiable cases for η=12 this implies that the Miyamoto involutions are 3-transpositions, leading to a classification.

Griess algebras generated by the Griess algebras of two $3A$-algebras with a common axis

In this article, we study Griess algebras generated by two pairs of Ising vectors $(a_0, a_1)$ and $(b_0,b_1)$ such that each pair generates a $3A$-algebra $U_{3A}$ and their intersection contains the $W_3$-algebra $\mathcal{W}(4/5)\cong L(4/5,0)\oplus L(4/5,3)$. We show that there are only 3 possibilities, up to isomorphisms and they are isomorphic to the Griess algebras of the VOAs $V_{F(1A)}$, $V_{F(2A)}$ and $V_{F(3A)}$ constructed by Hohn–Lam–Yamauchi.

An explicit Majorana representation of the group 32:2 of 3C-pure type

In this article, we study Griess algebras and vertex operator subalgebras generated by Ising vectors in a moonshine type VOA such that the subgroup generated by the corresponding Miyamoto involutions has the shape $3^2{:}2$ and any two Ising vectors generate a 3C subVOA $U_{3C}$. We show that such a Griess algebra is uniquely determined, up to isomorphisms. The structure of the corresponding vertex operator algebra is also discussed. In addition, we give a construction of such a VOA inside the lattice VOA $V_{E_8^3}$, which gives an explicit example for Majorana representations of the group $3^2{:}2$ of 3C-pure type.

Associative subalgebras of low-dimensional Majorana algebras

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.A. Ivanov in 2009 inspired by the work of S. Sakuma and M. Miyamoto. In the present paper, we revisit Mayer and Neutsch's theorem on associative subalgebras of the Griess algebra in the context of Majorana theory. We apply this result to determine all the maximal associative subalgebras of some low-dimensional Majorana algebras; namely, the Majorana algebras generated by two Majorana axes and the Majorana representations of the symmetric group of degree 4 involving 3C-algebras.

Weyl groups and vertex operator algebras generated by Ising vectors satisfying the (2B, 3C) condition

In this paper, we construct explicitly certain moonshine type vertex operator algebras generated by a set of Ising vectors I such that(1) for any e ≠ f ∈ I, the subVOA VOA (e, f) generated by e and f is isomorphic to either U2Bor U3C; and(2) the subgroup generated by the corresponding Miyamoto involutions {τe| e ∈ I} is isomorphic to the Weyl group of a root system of type An, Dn, E6, E7or E8.The structures of the corresponding vertex operator algebras and their Griess algebras are also studied. In particular, the central charge of these vertex operator algebras are determined.