3 -generated axial algebras with a minimal Miyamoto group
aa r X i v : . [ m a t h . R A ] A p r J. M c Inroy ∗ April 28, 2020
Abstract
Axial algebras are a recently introduced class of non-associative al-gebra, with a naturally associated group, which generalise the Griessalgebra and some key features of the moonshine VOA. Sakuma’s The-orem classifies the eight 2-generated axial algebras of Monster type. Inthis paper, we compute almost all the 3-generated axial algebras whoseassociated Miyamoto group is minimal 3-generated (this includes theminimal 3-generated algebras). We note that this work was carriedout independently to that of Mamontov, Staroletov and Whybrow andextends their result by computing more algebras and not assumingprimitivity, or an associating bilinear form.
There has been much interest recently in axial algebras and in particulartheir construction. There are eight 2-generated axial algebras of Monstertype and these are classified in Sakuma’s Theorem [1]. In this note, we lookat 3-generated axial algebras.Axial algebras are non-associative algebras which axiomatise some keyfeatures of VOAs and the Griess algebra. For full details, we refer read-ers to [1], or for a more general treatment [3]. Roughly speaking, they aregenerated by a set X of distinguished elements, called axes . The multi-plicative action of an axis on the algebra decomposes it into a direct sumof eigenspaces and the multiplication of these eigenspaces is governed bya so-called fusion law. This gives us partial control over multiplication inthe algebra. Importantly, when this fusion law is graded, we can associateautomorphisms, called Miyamoto automorphisms , to each axis. The groupgenerated by all of these is called the
Miyamoto group . ∗ School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol,BS8 1UG, UK; and the Heilbronn Institute for Mathematical Research, Bristol, UK; email:[email protected]
Monster type . Note that, since the Monster fusion law is Z -1 0
14 132
14 132
14 13214 14 14 , , , Table 1: Monster fusion lawgraded, associated to each axis a , there is at most one non-trivial Miyamotoautomorphism τ a and it has order 2.Sakuma’s theorem classifies the 2-generated axial algebras of Monstertype and shows that there are eight in total and they are labelled 2A, 2B, 3A,3C, 4A, 4B, 5A and 6A. Here the number indicates the order of the product τ a τ b in the Monster, where a and b generate the algebra. In particular, theorder of the product of any two Miyamoto involutions in an axial algebraof Monster type must have order at most 6. So, the Miyamoto group of anaxial algebra of Monster type is a 6-transposition group ( G, D ), where eachMiyamoto involution τ a ∈ D .In this note, we enumerate the axial algebras of Monster type whoseMiyamoto group G is minimal 3-generated. That is, where G is 3-generated,but any proper subgroup H (cid:8) G with H = h H ∩ D i can be generated by twoinvolutions in D . Note that we must impose some additional restriction ona 3-generated group G to make the problem tractable. Indeed, the Monsteris 3-generated and a 3-generated 5-transposition group B (2 ,
5) : 2 containsan open case of the Burnside problem. See [2] for an alternative restriction.We note that this work was carried out independently of that of Mamon-tov, Staroletov and Whybrow in [4]. However, both groups use the same listof minimal 3-generated 6-transposition groups which was originally given byMamontov and Staroletov [4, Theorem 2.2] and communicated to this au-thor by Shpectorov at the Focused Workshop on Axial Algebras in May/June2018.We record our results here as they differ from theirs in the followingways: • We do not assume that the algebra has a Frobenius form. That is, abilinear form which associates with the algebra product. (Indeed, Ma-2ontov, Staroletov and Whybrow assume a positive-definite Frobeniusform.) • We do not assume primitivity i.e. that the 1-eigenspace of every axisis 1-dimensional. • We complete some cases which they could not (in addition to all thosethat they can). Namely, we obtain two new non-trivial algebras andshow a further seven do not lead to non-trivial algebras. Furthermore,for one case we obtain a larger algebra which is a cover of the examplein [4].For the construction of the algebras, we use the magma implementa-tion [7] of the algorithm in [6], rather than the algorithm in [8] used byMamontov, Staroletov and Whybrow.
Theorem 1.1. If A is a -generated axial algebra of Monster type whoseMiyamoto group is minimal -generated, then it is a quotient of some algebrain Table . Note however that there are still twelve cases outstanding out of a totalof 161. Excluding these uncompleted cases and the two marked algebrascomputed over Q [ t ], we have the following: Corollary 1.2.
All the completed algebras in Table are primitive andadmit a positive semi-definite Frobenius form. This supports the conjecture in [6] that every axial algebra of Monstertype admits a Frobenius form.
Given a 3-generated 6-transposition group G = h x, y, z i , where D = x G ∪ y G ∪ z G , we describe briefly how to find all possible actions on a putativeset of axes X , so that ( A, X ) could be an axial algebra of Monster type withMiyamoto group G . For complete details, see Section 4 of [2].Using the action on X , we may assume that a, b, c are axes such that τ a = x , τ b = y and τ c = z and X = a G ∪ b G ∪ c G . (Note that we do notassume that these orbits are disjoint.) We begin by finding the possiblestabilisers G d of an axis d , for d = a, b, c . Note first that: h τ d i ≤ G d ≤ C G ( τ d ) Lemma 2.1. [2, Lemma 4.1] τ d = 1 if and only if G d = G . We may now assume that τ d is not the identity. To help identify G d , wemake the following definitions. 3 efinition 2.2.
1. An axis d has the uniqueness property if there doesnot exist another axis e ∈ X with d = e and τ d = τ e .2. An axis d is strong if there does not exist another axis e ∈ d G with d = e and τ d = τ e .Clearly an axis with the uniqueness property is strong. Strong axes haveaxis stabilisers which are as large as possible: Lemma 2.3. [2, Lemma 4.6]
The following are equivalent d is strong G d = C G ( τ d )3. There is a natural G -equivariant bijection between d G and τ Gd . With this in mind, we use the two following lemmas to gain a lowerbound for G d . Lemma 2.4. [2, Lemma 4.3]
Let d ∈ X . If there exists e ∈ X such that τ d τ e has order , then d has the uniqueness property. Lemma 2.5. [2, Lemma 4.7]
Let d, e ∈ X . If the order of τ d τ e is and e is strong, then τ e ∈ G d . If the order of τ d τ e is , then ( τ d τ e ) ∈ G d . For G = h x, y, z i , we enumerate all possible triples ( G a , G b , G c ) of axisstabilisers up to conjugation. This naturally gives us a triple of orbits( a G , b G , c G ), but we must still determine whether two orbits are disjointor not. If two of the generators τ d and τ e are conjugate and G d is conjugateto G e , then there are two cases: either d G and e G are equal, or disjoint.Otherwise, there is just one case and d G and e G are disjoint. Taking intoaccount all these different choices we may enumerate all possible actions of G on the putative set of axes X = a G ∪ b G ∪ c G .As described in [6], given a set of axes X , we may now construct theadmissible τ -maps. That is, those maps τ : X → G which could be a mapgiving the Miyamoto involutions τ x for each axis x ∈ X . In particular,Im( τ ) = G and τ xg = ( τ x ) g for all g ∈ G . Finally, for a given X and τ ,we determine the set of possible shapes, where a shape is an assignment of2-generated subalgebra to each orbit of pairs of distinct axes. This is theinformation needed for input into the algorithm. Not that at each stage herein determining the action, τ -map and shape, we do this up to appropriateautomorphisms. 4 Results
In this section we present our results in Table 2. The columns in this tableare • Miyamoto group. • Axes, where we give the size decomposed into the sum of orbit lengths. • Shape. If an algebra contains a 4A, or 4B, we omit to mention the2B, or 2A, respectively, which is contained in it. Similarly, we omitthe 2A and 3A contained in a 6A. We also usually omit the 6A and5A as these are uniquely defined. • Whether the example is a minimal 3-generated algebra in the sensethat it does not contain any 3-generated axial subalgebras. For thecases we could not complete, we indicate whether it could be minimalor not. • Dimension of the algebra. A question mark indicates that our algo-rithm did not complete and a 0 indicates that the algebra collapses. • The minimal m for which A is m -closed. Recall that an axial algebrais m -closed if it is spanned by products of length at most m in theaxes. • Whether the algebra has a Frobenius form that is non-zero on the set ofaxes X . If it is additionally positive definite or positive semi-definite,we mark this with a pos, or semi, respectively.We now comment on our results. Firstly, we complete all the casescompleted in [4] and some additional cases. In particular, we find two newnon-trivial axial algebras, show an additional seven cases collapse and find alarger algebra for another case (this contains the algebra in [4] as a quotient).(We also have a different result in one other case which we believe to be atypo in [4]. ∗ )The two new algebras are the 23-dimensional and 36-dimensional alge-bras for S on 3 + 6 + 6 axes with shapes (4A) (2A) and (4A)
2A 2B 2A,respectively. The cases which collapse are 2 on 2 + 2 + 4 axes with shape(4A) A A B ; 2 on 4+4+4 axes with shapes (4A) (2A) , (4A) (2A)
2A (2B) , (4A) (4B) (2A) and (4A) (4B)
2A 2B 2A and S on 6 +6 + 6 axes with shape (2A) . Interestingly, one of the two new algebrasis 3-closed, but the other is 2-closed and other 3-closed algebras were con-structed in [4], so the difference between the two algorithms is not due tothis. ∗ We find that S on 1 + 3 + 3 axes with shape 6A (2A) has dimension 8 and not 11.Indeed, it is isomorphic to the 6A Norton-Sakuma algebra.
5n addition, we find a 16-dimensional algebra A for 2 on 2 + 4 + 4 axeswith shape (4B) (4A) (2A) , whereas in [4] this is listed as 13-dimensional.However, our algebra A has a positive semi-definite Frobenius form andwhen you factor out by the 3-dimensional radical of this form, we recover a13-dimensional algebra with the same shape. We believe that this accountsthe difference here and confirms that Mamontov, Staroletov and Whybrowdo indeed assume a positive definite Frobenius form.For the constructed non-trivial algebras, it is easy to check that they areall primitive and admit a Frobenius form, giving us Corollary 1.2. G axes shape minimal dim m form1 1+1+1 (2A) yes 6,9 2,3 pos1 1+1+1 (2A)
2B yes 6 3 pos1 1+1+1 2A (2B) yes 4 2 pos1 1+1+1 (2B) yes 3 1 pos2 no 14 3 semi2 no 6 2 pos2 yes 5 1 pos2 no 6 2 pos2 (2A) no 0,10,? †
2A 2B - 0 0 -2 (2B) no 9 3 pos2 (2A) no 11 2 pos2
2A 2B no 8 2 pos2 (2B) - 0 0 -2 yes 12 ‡ yes 7 2 pos S S S S † Assuming primitivity, there are three possible cases when the algebra is evaluatedover Q [ t ]. For all but two values of t , the algebra collapses. When t = , the algebra is10-dimensional and it does not complete if t = − . For more details, see [5] ‡ Assuming primitivity, there is a 12 dim example over Q [ t ]. When evaluated for aparticular value of t , the example always admits a Frobenius form, but it can be positivedefinite, positive semi-definite, or not either depending upon the value of t . This was firstconstructed by Whybrow in [9]. no 8 ∗ S S no 9 2 pos S no 13 2 pos2 (2A) - 0 0 -2 A A B - 0 0 -2 A B A - 0 0 -2 A B B no ?2 B B A - 0 0 -2 (2B) no 13 3 pos2 - 0 0 -2 A A B - 0 0 -2 A B A no 15 3 pos2 A B B - 0 0 -2 B B A no 12 2 pos2 - 0 0 -2 (2A) - 0 0 -2 A A B no ?2 A B A - 0 0 -2 A B B - 0 0 -2 B B A - 0 0 -2 (2B) no 10 2 pos2
2A - 0 0 -2
2B - 0 0 -2
2A - 0 0 -2
2B - 0 0 -2 (2A) - 0 0 -2
2A 2B - 0 0 -2 (2B) - 0 0 -2 (4B) (2A) - 0 0 -2 (4B)
2A 2B - 0 0 -2 (4B) (2B) - 0 0 -2 (4A) (2A) no 16 2 semi2 (4A)
2A 2B - 0 0 -2 (4A) (2B) - 0 0 -2 (2A) - 0 0 -2
2A 2B - 0 0 -2 (2B) - 0 0 -2 (2A) - 0 0 -7 (2A)
2B - 0 0 -2
2A (2B) - 0 0 -2 (2B) no ?2 (4B) (2A) - 0 0 -2 (4B)
2A 2A 2B no ?2 (4B)
2A 2B 2A - 0 0 -2 (4B)
2A 2B 2B - 0 0 -2 (4B)
2B 2B 2A - 0 0 -2 (4B) (2B) - 0 0 -2 (4B) (2A) - 0 0 -2 (4B)
2A 2A 2B - 0 0 -2 (4B)
2A 2B 2B - 0 0 -2 (4B)
2B 2A 2A no ?2 (4B)
2B 2A 2B - 0 0 -2 (4B) (2B) - 0 0 -2 (2A) - 0 0 -2 (2A)
2B - 0 0 -2
2A (2B) - 0 0 -2 (2B) no 15 2 pos D D D D D D - 0 0 - D
2A - 0 0 - D
2B - 0 0 -3 : 2 9 (3A) yes ?3 : 2 9 (3A)
3C - 0 0 -3 : 2 9 (3A) (3C) yes ?3 : 2 9 3A (3C) yes 12 2 pos3 : 2 9 (3C) yes 9 1 pos3 : 2 9+9 (6A) no 31 2 pos3 : 2 9+9+9 (6A) no 42 2 pos S S S S S S S S S S S S
12 6A yes 17 2 pos S - 0 0 - S
2B no 20 2 pos S S - 0 0 - S
2A no ? S - 0 0 - S (2A) no 23 2 pos S
2A 2A 2B - 0 0 - S
2A 2B 2A no 36 3 pos S
2A 2B 2B - 0 0 - S
2B 2A 2B no ? S (2B) - 0 0 - S (2A) - 0 0 - S
2A 2A 2B no 20 2 pos S
2A 2B 2A - 0 0 - S
2A 2B 2B - 0 0 - S
2B 2A 2B - 0 0 - S (2B) no 23 2 pos S - 0 0 - S
2A 2B - 0 0 - S
2B 2B - 0 0 - S
2B 2A 2A - 0 0 - S
2B 2A 2B - 0 0 - S
2B 2B 2A - 0 0 - S
2B (2B) - 0 0 - S
2B 2A (2B) - 0 0 - S (2B) (2A) - 0 0 -9 (2B)
2A 2B - 0 0 - S (2B)
2B 2A - 0 0 - S (2B) (2B) - 0 0 - S (2B) no 28 2 pos S - 0 0 - S
2A 2B - 0 0 - S - 0 0 - S (2A) no ? S
2A 2B - 0 0 - S (2B) - 0 0 - S - 0 0 - S - 0 0 - S - 0 0 -5 : 2 25 (5A) yes ?3 : S
27 (6A) - 0 0 -3 : S - 0 0 -3 : S - 0 0 -3 : S - 0 0 -3 : S - 0 0 -3 : S - 0 0 - A
15 3A 2A yes 26 2 pos A
15 3A 2B no 46 3 pos A
15 3C 2A yes 20 2 pos A
15 3C 2B no 21 2 pos5 : S
15 3A yes ?5 : S
15 3C yes 18 2 pos
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