Research topics in finite groups and vertex algebras

TL;DR

This paper proposes research projects focusing on vertex algebras through the lens of finite group theory, aiming to deepen understanding in this mathematical area.

Contribution

It introduces new research directions connecting finite groups and vertex algebras, highlighting potential for future exploration.

Findings

Identified key research projects in vertex algebras

Emphasized finite group perspectives in the study of vertex algebras

Suggested avenues for advancing the field

Abstract

We suggest a few projects for studying vertex algebras with emphasis on finite group viewpoints.

Full text

1. 1 Introduction

2. 1.1 A few definitions

3. 2 Derivations

4. 3 The $\omega$-transposition property

5. 4 Automorphism groups

6. 5 Baer-Suzuki theorem for MIs

7. 6 Real, complex and quaternion reflection groups

8. 7 Fixed point subVOAs

9. 7.1 An example

10. 8 Integral form classifications for $V$ and $V_{2}$

11. 9 Uniqueness of dihedral VOA

12. 10 Integral Forms

13. 11 Module theory for Chevalley-Steinberg groups

14. 12 Acknowledgments

1 Introduction

In this article, we suggest a few projects for studying VOAs with emphasis on finite group viewpoints. The world of VOAs is quite large and mostly unexplored. It may be worthwhile to cultivate connections between the theories of finite groups and VOAs since many interesting VOAs are associated to finite groups.

Some familiar VOAs, such as lattice type, have been extensively studied, both by Lie theoretic and discrete methods. In contrast, the Monster VOA has no nonzero derivations [18] and much of its study involves properties of its finite automorphism group, the Monster.

We will list a series of topics which seem worthwhile for research. In a sense, this article is a successor to [22].

1.1 A few definitions

For basic background on vertex operators, see [18]. An Ising vector in a VOA is a conformal vector of central charge $\frac{1}{2}$ which generates a simple subVOA; it defines a Miyamoto involution (MI) on the VOA [43]. A VOA has CFT type if it has the form $\bigoplus_{n\geq 0}V_{n}$ so that $dim(V_{0})=1$. A VOA has OZ type (or is an OZVOA) if it has CFT type and $V_{1}=0$ [22] (“OZ” stands for one-dimensional, zero-dimensional). Integral forms (IF) for a VOA are defined in [13].

Group theoretic notation will be generally consistent with [21, 26].

2 Derivations

For background, see [12].

Topic 2.1**.**

Find general conditions on a VOA which make every derivation inner.

Such a result would be an analogue of the property that every derivation is inner for finite dimensional complex semisimple Lie algebras.

Topic 2.2**.**

Find general conditions on a VOA which make a VOA have non-inner derivations. Furthermore, find general upper and lower bounds on the dimension of the space of derivations in terms of dimensions of finite dimensional homogeneous subspaces which generate.

Results of this type might be analogous to results for finite groups about the size of the outer automorphism group. The latter issue can involve first cohomology groups [29]. See also the Gäschutz theorem that most finite $p$-groups have nontrivial outer automorphism group [20] and p.403 [33].

3 The $\omega$-transposition property

Topic 3.1**.**

Find analogues in VOA theory of classifications in finite group theory by special generators, meaning the subgroups generated by two such elements are restricted. See [16, 17, 50, 49, 34, 35, 36, 37, 38, 39, 40, 2, 3, 4]

The Miyamoto Involutions [43] are among the most accessible automorphisms of VOAs. They do not exist for every VOA but they do exist in many cases where the VOA has finite automorphism groups. In case the MIs have $\sigma$-type, they satisfy the famous 3-transposition condition of Fischer. In case the MIs have $\tau$-type, and the VOA has a real form with positive definite natural bilinear form [46], the MIs satisfy a 6-transposition property (this is the case for $2A$ involution of the Monster action on the Monster VOA).

Topic 3.2**.**

Define other general types of involutions on families of VOAs and prove that they satisfy some limited set of orders of products of pairs.

See [31, 32, 42] for related results.

4 Automorphism groups

Topic 4.1**.**

Given a finite group $G$, is there a VOA whose automorphism group is $G$, or essentially $G$? Better: is there such a VOA of CFT type?

The meaning of “essentially” here is $G$ occurs as $H/K$, where $H,K$ are normal subgroups of $Aut(V)$, for some VOA $V$, and where $K$ and $Aut(V)/H$ are solvable and are small relative to $G$. The first example of this is probably the case where $G$ is the Monster; the Moonshine VOA has $G$ as its automorphism group [18]. There are recent results for $G\cong Sym_{n}$ [41].

If $V$ is a VOA of CFT and $V_{1}\neq 0$, the (infinite) complex Lie group associated to the Lie algebra $V_{1}$ acts on $V$ as automorphisms, though not always faithfully. An example is the Heisenberg VOA $M(1)$ of central charge 1, whose automorphism group is cyclic of order 2. Rationality may be relevant [12].

Topic 4.2**.**

Same as the previous topic except replace VOA over the complex numbers by a VA over a finite field.

The answer is positive for the series of Chevalley-Steinberg groups defined over a given finite field [25]. The answer is unknown for almost all of the remaining families for groups of Lie type, the Suzuki groups $Sz(q)={}^{2}B_{2}(q)$ for $q\geq 8$ an odd power of $2$ and for the Ree groups ${}^{2}F_{4}(q)^{\prime}$ for $q$ an odd power of 2 (note commutator subgroup at $q=2$) and ${}^{2}G_{2}(q)$ for $q\geq 27$ an odd power of 3 (${}^{2}G_{2}(3)\cong SL(2,8):3$).

Topic 4.3**.**

The twisted lattice type VOA for Barnes-Wall lattice of rank 32 has automorphism group of the form $2^{27}.E_{6}(2)$ [47]. This group has the form of a parabolic subgroup of $E_{7}(2)$. Find VOAs whose automorphism groups are essentially $E_{6}(2),E_{7}(2),E_{8}(2)$ and some of their parabolic subgroups.

If $V$ is the VOA of [47], $Aut(V)\cong 2^{27}{:}E_{6}(2)$. Let $A:=O_{2}(Aut(V))\cong 2^{27}$, then we have an action of $E_{6}(2)$ as automorphisms of $V^{A}$. Possibly, $Aut(V^{A})\cong E_{6}(2)$ or $E_{6}(2){:}2$. See [28] and Example 3.2 of [22].

5 Baer-Suzuki theorem for MIs

The Baer-Suzuki theorem says that in a finite group, $G$, if $p$ is a prime number and $x\in G$ has order a power of $p$, then $x\in O_{p}(G)$ if and only if $\langle x,y\rangle$ is a $p$-subgroup for all conjugates $y$ of $x$. See [21, 1].

We generalize a bit with this hypothesis: $p$ is a prime number, $K,L$ are conjugacy classes in the finite group $G$ with the property that $\langle x,y\rangle$ is a $p$-group, for all $x\in K,y\in L$.

The Baer-Suzuki theorem says that if $K=L$, $K$ is contained in $O_{p}(G)$. If $K\neq L$, $K$ may not be contained in $O_{p}(G)$.

For example, in $Sym_{6}$, take $p=2$ we let $K$ be the conjugacy class of $(12)$ and let $L$ be the conjugacy class of $(12)(34)(56)$. Then $\langle x,y\rangle$ is a 2-group for all $x\in K,y\in L$. There may be counterexamples for $p=3$ in $G_{2}(3)$, but we do not know a reference.

Topic 5.1**.**

Is there a proof by VOA theory that the conclusion of the Baer-Suzuki theorem holds for a conjugacy class of Miyamoto involutions?

Topic 5.2**.**

Is there an analogue of the Baer-Suzuki theorem which might be applicable to (short) words in MIs?

6 Real, complex and quaternion reflection groups

Finite real and complex reflection groups have been classified and studied a long time ago. There is a less well-known classification of quaternionic reflection groups [9], whose list of conclusions includes double covers of alternating groups and the double cover of the Hall-Janko group.

Topic 6.1**.**

Is there a class of VOAs which are quaternionic vector spaces and in which there a class of automorphisms of VOAs which behave like quaternionic reflections? Are there such VOAs which realize the finite groups generated by quaternionic reflections groups as their automorphism groups.

7 Fixed point subVOAs

Topic 7.1**.**

Start with $V$ a familiar VOA, $F$ a finite subgroup of $Aut(V)$ and study $V^{F}$. Determine properties of $V^{F}$, such as automorphism group, $C_{2}$, regularity, etc. Decide whether $V^{F}$ is isomorphic to a familiar VOA. One may continue by taking a finite subgroup $S$ of $Aut(V^{F})$, then study $(V^{F})^{H}$, etc.

Parts of the above program were done for $V=V_{A_{1}}$ [10, 11]. See also the recent [7, 31, 32, 42]. The cases $V=V_{L}$ where $L$ is a root lattice of type ADE are likely to be interesting. There are many cases of finite group $F$ in $Aut(V)$ to try. See the survey [26] and [27]. Note that there are examples in [26] for which $V_{1}^{F}=0$. This means that $V^{F}$ is an OZVOA and in particular $V_{2}^{F}$ is a commutative algebra.

7.1 An example

Let $V=V_{E_{8}}$ and let $F$ be the Borovik group, i.e. a subgroup $B$ of $Aut(V)\cong E_{8}({\mathbb{C}})$ which has these properties:

(i) $B$ contains a normal subgroup $B_{5}\times B_{6}$, where $B_{k}\cong Alt_{k}$ and $C_{B}(B_{5})=B_{6}$ and $C_{B}(B_{6})=B_{5}$;

(ii) By conjugation on $B_{5}\times B_{6}$, $B$ embeds as an index 2 subgroup of $Aut(B_{5})\times Aut(B_{6})$;

(iii) $B$ contains a subgroup isomorphic to $Alt_{5}\times Sym_{6}$; and

(iv) $B$ is not a nontrivial direct product.

This is a Lie primitive subgroup of $E_{8}({\mathbb{C}})$ (i.e., not contained in a positive dimensional Zariski-closed subgroup except for $E_{8}({\mathbb{C}})$). See [5, 8] for characterization of $B$ and [19] for the $E_{8}({\mathbb{C}})$-conjugacy classes represented in $B$.

In particular, $V_{1}^{B}=0$, $dim(V_{2}^{B_{5}})=16$ and $dim(V_{2}^{B_{6}})=10$ [22] . Since $B/B_{5}\cong Aut(Alt_{6})$ acts as automorphisms on the commutative algebra $V_{2}^{B_{5}}$, a reasonable guess would be that $V_{2}^{B_{5}}\cong Mat_{4\times 4}^{+}$, a Jordan algebra. Since $B/B_{6}\cong Sym_{5}$ acts as automorphisms on the commutative algebra $V_{2}^{B_{6}}$, a reasonable guess would be that $V_{2}^{B_{5}}\cong SymMat_{4\times 4}$, a Jordan algebra.

Topic 7.2**.**

Identify the fixed point subalgebras $V_{2}^{B_{5}}$ and $V_{2}^{B_{6}}$. Determine their automorphism groups.

Topic 7.3**.**

For $F=B_{5}$ or $B_{6}$, let $V_{2}^{F}$ be the fixed point subalgebra in $V_{2}$. Find Ising vectors (conformal vectors of central charge 1/2) in $V_{2}$ whose Miyamoto involutions give the $2A$-involutions of $B$; see [19].

We remark that the function which sends an Ising vector to its Miyamoto involution is in general not a monomorphism [24]. Given a $2A$-involution of $B$, it is possible that some Ising vectors which represent it are in $V_{2}^{F}$, or that none are.

8 Integral form classifications for $V$ and $V_{2}$

Studies of integral forms in VOAs are not (yet) numerous. See [23] for a survey. Most results are existence proofs.

Topic 8.1**.**

Give existence proofs of integral forms for more VOAs of interest, and give uniqueness proofs (say for maximal integral forms).

When the automorphism group of a VOA is a positive dimensional algebraic group, an integral form has an infinite orbit under the automorphism group, so can not be unique. A uniqueness program must be sensitive to the fact that different integral forms may have different signatures (as real quadratic forms).

Topic 8.2**.**

Determine a suitable notion of equivalence for group-invariant integral forms (or maximal integral forms) in a given VOA, and give conditions under which integral forms have finitely many equivalence classes (and even one equivalence class).

A step in this direction was obtained in the thesis of Simon [48], who classified maximal integral forms in $S$ which are invariant under certain finite groups. Here, $S$ is one of the nine dihedral algebras determined by Sakuma [46]. These algebras are naturally labeled by nodes of the extended $E_{8}$-diagrams. This $S$ occurs as $V_{2}$, where $V$ is a VOA generated by a pair of Ising vectors corresponding to one of the nine configurations in [46].

Topic 8.3**.**

Determine the maximal, group-invariant integral forms in a given dihedral VOA, $V$. Is it true that such an integral form is generated by one of the integral forms in $V_{2}$ as classified in [48]?

9 Uniqueness of dihedral VOA

Suppose that the VOA $V$ has OZ type.

Topic 9.1**.**

To what extent is $V$ is determined by its degree 2 summand, $V_{2}$?

A necessary condition for $V$ to be uniquely determined by $V_{2}$ is that it be generated by its degree 2 summand.

Topic 9.2**.**

When $V_{2}$ is one of the nine Sakuma dihedral algebras and $V_{2}$ generates $V$, is $V$ unique?

The answer is yes for at lease one case of the Sakuma dihedral algebras. See [15].

The Sakuma classification [46] follows from a more general later result [30].

10 Integral Forms

Existence proofs for integral forms in VOAs have begun to appear in recent literature [23]. Classifications of integral forms are fewer. The thesis of Simon [48] classifies certain maximal group-invariant integral forms in the nine degree 2 Sakuma dihedral algebras, and in a few other commutative nonassociative algebras.

Topic 10.1**.**

Extend the above classifications of IFs in Sakuma algebras to IFs in full dihedral VOAs.

Topic 10.2**.**

Suppose that the $V$ is holomorphic. Is there self-dual integral form in $V$ unique, up to equivalence. In particular, does the Moonshine VOA have a unique self-dual integral form?

In [6], Carnahan proposes four integral self-dual forms for the Moonshine VOA and conjectures that they ought to be equal.

11 Module theory for Chevalley-Steinberg groups

Let $\mathbb{F}\leq\mathbb{E}$ be a finite degree field extension and $\Phi$ an indecomposable root system. In [25], Griess and Lam gave constructions for classical VAs of type $\Phi$ over $\mathbb{E}$ and of the related Chevalley-Steinberg groups associated to $\mathbb{F}\leq\mathbb{E}$. These groups are essentially the full automorphism groups for these VAs.

For the positive characteristic case, the finite dimensional modules for these groups are generally not completely reducible.

Topic 11.1**.**

For each graded piece of the VA, determine the composition factors with respect to the associated Chevalley-Steinberg group and determine a direct sum decomposition into indecomposable modules.

Topic 11.2**.**

By examining the graded pieces of the VA, deduce lower bounds for dimensions of $Ext^{n}(A,B)$ for nonzero modules $A,B$ for the automorphism group and for its socle.

Constructing nonsplit extensions of one module by another can be technically difficult. The point here is that the VA graded space gives at once a vast collection of modules to study.

12 Acknowledgments

Our talk given at the Mason fest 2018, a survey of integral form results in vertex operator algebras, is represented in articles already committed to other publications [23, 14].

Work on the present article was supported by funds from University of Michigan.

We thank the referee for comments and references.