Associative Subalgebras of the Norton-Sakuma Algebras A. Castillo-Ramirez∗ Imperial College London, Department of Mathematics. South Kensington Campus, London, SW7 2AZ. Email address:
[email protected]
October 8, 2013
Abstract A Majorana representation of a transposition group is a non-associative commutative real algebra that satisfies some of the properties of the Griess algebra. The term was introduced by A. A. Ivanov in 2009 inspired by Sakuma’s theorem, which establishes that the Majorana representations of the dihedral groups are the so-called Norton-Sakuma algebras. Since these algebras classify the isomorphism types of any algebra generated by two Majorana axes, they have become the fundamental building blocks in the construction of Majorana representations. In the present paper, we revisit Mayer and Neutsch’s theorem on associative subalgebras of the Griess algebra in the context of Majorana theory, and we apply this result to determine all the maximal associative subalgebras of the NortonSakuma algebras.
∗ Funded by the Universidad de Guadalajara and an Imperial College International Scholarship.
1
Majorana Representations
The largest of the sporadic simple groups, the Monster gorup M, was constructed for the fist time by Griess [G82] as a group of automorphisms of a 196, 884-dimensional commutative nonassociative algebra VM . This construction was later simplified by Conway [C84], who, among various things, defined idempotents ψ(t) ∈ VM , called 2A-axis, associated with every 2A-involution t ∈ M. Frenkel, Lepowsky and Meurman [FLM] constructed a Vertex Operator Algebra (VOA) V \ , called the Moonshine module, with the properties that Aut(V \ ) = M and the weight 2 subspace of V \ coincides with VM . Let ψ(t) and ψ(g) be distinct 2A-axes of VM . Norton [N96] showed that the possible isomorphism types of the subalgebra of VM generated by ψ(t) and ψ(g) may be labeled by 2A, 2B, 3A, 3C, 4A, 4B, 5A and 6A, as they are completely determined by the conjugacy class in M of the product tg. In the context of a generalized Griess subalgebra of a VOA, Sakuma [Sa07] showed that the possible isomorphism types of an algebra generated by two distinct Ising vectors coincides with the isomorphism types described by Norton. Inspired in Sakuma’s theorem, Ivanov [Iv09] defined the concept of a Majorana representation of a transposition group; before stating its definition we introduce some notation. Let V = (V, ·, (, )) be a commutative (not necessarily associative) real algebra with inner product. For S ⊆ V , denote by hhSii the smallest subalgebra of V containing S. Define the adjoint transformation of v ∈ V as the linear map adv (u) = v · u for u ∈ V. We say that µ ∈ R is an eigenvalue of v ∈ V if µ is an eigenvalue of the (v) transformation adv . In this situation, denote by Vµ the µ-eigenspace of adv . Definition 1. A pair (G, T ) is called a transposition group if G is a finite group and T is a G-stable set of involutions of G such that hT i = G. Definition 2. Let (G, T ) be a transposition group, V = (V, ·, (, )) a commutative real algebra with inner product, ϕ : G → GL(V ) a linear representation of G such that ϕ(G) ≤ Aut(V ) and ψ : T → V \ {0} an injective map such that ϕ(g)
ψ (tg ) = ψ (t)
for all t ∈ T, g ∈ G.
The quintuple (G, T, V, ϕ, ψ) is called a Majorana representation of (G, T ) if the following M1-M8 axioms are satisfied: M1 The inner product associates with the algebra product in the sense that (u, v · w) = (u · v, w) for all u, v, w ∈ V. M2 For every u, v ∈ V , the Norton inequality holds: (u · u, v · v) ≥ (u · v, u · v). 2
M3 The elements of ψ (T ) are idempotents of length 1 such that hhψ(T )ii = V . M4 The elements of ψ (T ) are semisimple with spectrum contained in 1 1 Sp := 0, 1, 2 , 5 . 2 2 M5 For any a ∈ ψ (T ), 1 is a simple eigenvalue of a. M6 For any a ∈ ψ (T ), the endomorphism τ (a) of V defined by 5
uτ (a) := (−1)µ2 v, for v ∈ Vµ(a) , µ ∈ Sp, preserves the algebra product of V . M7 For any a ∈ ψ (T ), the endomorphism σ(a) of CV (τ (a)) defined by 1 σ(a) µ22 (a) , u := (−1) v, for v ∈ Vµ , µ ∈ Sp \ 25 preserves the algebra product of CV (τ (a)). M8 For any t ∈ T , we have that τ (ψ (t)) = ϕ (t) . The idempotents of ψ (T ) are called Majorana axes while the automorphisms of ϕ (T ) are called Majorana involutions. In order to simplify notation, denote by at the Majorana axis corresponding to t ∈ T . In the language of VOAs, the Majorana involutions correspond to restrictions of the Miyamoto involutions [Mi96]. When ϕ and ψ are clear in the context, we simply say that V is a Majorana representation of (G, T ). Sakuma’s theorem implies that the Majorana representations of the dihedral groups coincide with the subalgebras of VM described by Norton; hence, these algebras have been named the Norton-Sakuma algebras. Besides this, Majorana representations of various transposition groups have been described in [IPSS10], [IS12], [Iv11b], [Iv11a] and [IS]. Meyer and Neutsch [MN93] showed the existence of a maximal 48-dimensional subalgebra of VM generated by 2A-axes by proving that every associative subalgebra of VM is generated by a set of pairwise orthogonal idempotents. Furthermore, they conjectured that 48 was the largest possible dimension of an associative subalgebra of VM . This conjecture was proved later by Miyamoto [Mi96]. In this paper, we discuss Meyer and Neutsch’s results in the context of an abstract Majorana representation. Furthermore, we find all the maximal associative subalgebras of the Norton-Sakuma algebras using the information of idempotents obtained in [CR13]. Some of the idempotents and maximal associative subalgebras described here had been found earlier in [LYY05] in the context of VOAs. The present paper enhances the published information about idempotents and completes the classification of the maximal associative subalgebras of the Norton-Sakuma algebras.
3
2
Associative subalgebras of Majorana representations
Throughout this section, we assume that V is a Majorana representation of (G, T ) with identity id ∈ V . We will not use explicitly axioms M6–M8 of Definition 2, but we will replace M2 by a stronger statement: M2’ The Norton inequality holds for every u, v ∈ V , with equality precisely when the adjoint transformations adu and adv commute. It is shown in Section 15 of [C84] that axiom M2’ holds in VM . The importance of this stronger axiom in our discussion lies in the following proposition: (x)
Lemma 2.1. Let x ∈ V be an idempotent. Then the eigenspaces V0 are subalgebras of V .
(x)
and V1
(x)
Proof. Let µ ∈ {0, 1} and y1 , y2 ∈ Vµ . Observe that, for i ∈ {1, 2}, (x · x, yi · yi ) = (x · yi , yi ) = (µyi , yi ) = (µyi , µyi ) = (x · yi , x · yi ). Therefore, M2’ implies that adx and adyi commute, and so x · (y1 · y2 ) = y1 · (x · y2 ) = µ(y1 · y2 ). (x)
This shows that (y1 · y2 ) ∈ Vµ . If v ∈ V , define the length of v by the non-negative real number l(v) := (v, v). The following are further useful elementary results. Lemma 2.2. Let {xi ∈ V : 1 ≤ i ≤ k} be a finite set of idempotents. Let λi ∈ R and suppose that k X x= λ i xi , i=1
is also an idempotent. Then, we have that l(x) =
k X
λi l(xi ).
i=1
Proof. The lemma follows by axiom M1 and the linearity of the inner product: l(x) = (x, x) = (x · x, id) = (x, id) =
k X i=1
λi (xi , id) =
k X
λi l(xi ).
i=1
Lemma 2.3. Let x, y ∈ V be idempotents. The following are equivalent: (i) (x, y) = 0. (ii) x · y = 0. (iii) x + y is idempotent. 4
Proof. If (x, y) = 0, Norton inequality implies that (x · y, x · y) ≤ (x · x, y · y) = (x, y) = 0. Hence x · y = 0 by the positive definiteness of the inner product. It is clear that statement (ii) implies (iii). Finally, Lemma 2.2 shows that (iii) implies (i). We say that two idempotents are orthogonal if they satisfy the equivalent (v) statements (i) - (iii) of Lemma 2.3. For v ∈ V , define d(v) := dim(V0 ). Lemma 2.4. Let x ∈ V be a non-zero non-identity idempotent. The following statements hold: (i) 1 and 0 are eigenvalues of x. (ii) For every g ∈ Aut(V ), the spectrum of xg is equal to the spectrum of x. (iii) If v ∈ V is a λ-eigenvector of x, then v is a (1 − λ)-eigenvector of id − x. (x)
(iv) If V0 has finitely many idempotents, then x is orthogonal to at most 2d(x) idempotents. Proof. Part (i) follows because x and id − x are 1- and 0-eigenvectors of adx , respectively. Part (ii) follows since, for any g ∈ Aut(V ), v is an eigenvector of adx if and only if v g is an eigenvector of adxg . Parts (iii) is trivial while part (iv) follows by Lemma 2.1 and B´ezout’s theorem. When x ∈ V is a non-zero non-identity idempotent, it is clear that {x, id−x} is the orthogonal basis of a 2-dimensional associative subalgebra Vx := hhx, id − xii ≤ V. Define V0 := hh0ii and Vid := hhidii. Definition 3. We say that an associative subalgebra U of V is trivial associative if U = Vx for some idempotent x ∈ V . An idempotent is decomposable if it may be expressed as a sum of at least two nonzero idempotents; otherwise, we say the idempotent is indecomposable. The following results were obtained in [MN93] in the context of the Griess algebra; however, it should be noted that they hold as well in the context of a Majorana representation with identity that satisfies axiom M2’. Proposition 2.5 (Meyer, Neutsch). An idempotent x of V is indecomposable if and only if 1 is a simple eigenvalue of x. Corollary 2.6. The Majorana axes of V are indecomposable. Theorem 2.7 (Meyer, Neutsch). Let U be a subalgebra of V . The following statements hold: (i) U is associative if and only if U has an orthogonal basis of idempotents. (ii) U is maximal associative if and only if id ∈ U and the idempotents in the orthogonal basis of U are indecomposable.
5
Corollary 2.8. A trivial associative subalgebra Vx ≤ V is maximal associative if and only if 0 and 1 are simple eigenvalues of x ∈ V . Proof. The result follows by Lemma 2.4 (iii) and Theorem 2.7 (ii). (x)
(x)
In view of Theorem 2.7, it is relevant to study the eigensapces V0 and V1 , where x ∈ V is an idempotent. The following lemmas will be useful in our discussion about the associative subalgebras of the Norton-Sakuma algebras. (x)
Lemma 2.9. Suppose that, for every idempotent x ∈ V , the space V0 has finitely many idempotents and d(x) ≤ 2. Then every associative subalgebra of V is at most three-dimensional. Proof. If {x, y, z, w} is a set of four pairwise idempotents of V , then x is orthogonal to 7 idempotents: y, z, w, y + z, y + w, z + w and y + z + w. This contradicts Lemma 2.4 (iv). (x)
Lemma 2.10. Let x ∈ V be an idempotent and suppose that V0 many idempotents. The following statements hold:
has finitely
(i) If d(x) = 1, then x is not contained in the orthogonal basis of any threedimensional associative subalgebra of V . (ii) If d(x) ≥ 2, then x is in the orthogonal basis of at most 2d(x)−1 − 1 threedimensional maximal associative subalgebras of V . Proof. Part (i) is trivial. Let d(x) ≥ 2 and suppose that {x, y, z} is the or(x) thogonal basis of a maximal associative subalgebra of V , where y, z ∈ V0 are (x) idempotents. Note that, if there is an idempotent w ∈ V0 such that {x, y, w} is the orthogonal basis of a maximal associative subalgebra, then Theorem 2.7 (ii) implies that x + y + z = id = x + y + w, so z = w. This shows that the three-dimensional maximal associative subalgebras of V with x in their orthogonal basis correspond to disjoint two-sets of (x) (x) non-zero idempotents of V0 . By Be´zout’s theorem, V0 has at most 2d(x) d(x) idempotents. Therefore, there are at most 2 2 −2 disjoint two-sets of non-zero (x) idempotents in V0 .
3
The Norton-Sakuma Algebras
Sakuma’s theorem [Sa07] states that the product of any two Majorana involutions is at most 6 and that there are at most eight possibilities for the isomorphism type of an algebra generated by two Majorana axes. The following version of this theorem was established in [IPSS10]. Theorem 3.1. Let (G, T, V, ϕ, ψ) be a Majorana representation. Let t, g ∈ T , t 6= g, and define ρ := ϕ(tg). For i ∈ Z, let agi := ψ tρi . Then the subalgebra hhat , ag ii of V is isomorphic to a Norton-Sakuma algebra of type N X, as described in Table 1, where N = |ρ|, X ∈ {A, B, C}.
6
Type Basis
Products
2A
at · ag =
at , ag , aρ
1 23 (at
+ ag − aρ ), at · aρ =
(at , ag ) = (at , aρ ) = (ag , aρ ) =
at , ag
at · ag = 0, (at , ag ) = 0
3A
at , ag , ag−1 , uρ
at · ag =
1 25 (2at
+ 2ag + ag−1 ) −
at · uρ =
1 32 (2at
− ag − ag−1 ) +
13 28 ,
33 5 211 uρ ,
5 25 uρ ,
uρ · uρ = uρ ,
(uρ , uρ ) =
23 5
+ ag − ag−1 ), (at , ag ) =
1 26
(at , uρ ) =
1 22 ,
+ aρ − ag ),
1 23
2B
(at , ag ) =
1 23 (at
3C
at , ag , ag−1
at · ag =
1 26 (at
4A
at , ag , ag−1 , ag2 , vρ
at · ag =
1 26 (3at
+ 3ag + ag2 + ag−1 − 3vρ ),
at · vρ =
1 24 (5at
− 2ag − ag2 − 2ag−1 + 3vρ ),
vρ · vρ = vρ , at · ag2 = 0, (at , ag ) = (at , ag2 ) = 0, (at , vρ ) =
4B
5A
6A
at , ag , ag−1 , ag2 , aρ2
1 26 (at
at · ag =
3 23 ,
(vρ , vρ ) = 2
+ ag − ag−1 − ag2 + aρ2 ),
at · ag2 =
1 23 (at
(at , ag ) =
1 26 ,
+ ag2 − aρ2 ),
(at , ag2 ) = (at , aρ2 ) =
1 27 (3at
at , ag , ag−1 , ag2 ,
at · ag =
ag−2 , wρ
at · ag2 =
1 27 (3at
at · wρ =
7 212 (ag
+ 3ag2 − ag1 − ag−1 − ag−2 ) − wρ , + ag−1 − ag2 − ag−2 ) +
52 ·7 219 (ag−2
(at , ag ) =
3 27 ,
(at , wρ ) = 0, (wρ , wρ ) =
1 26 (at + ag − ag−2
53 7 219
2
− ag−1 − ag2 − ag3 + aρ3 ) + 32115 uρ2 ,
at · ag =
ag−2 , ag3 , aρ3 , uρ2
at · ag2 =
1 25 (2at
+ 2ag2 + ag−2 ) −
at · uρ2 =
1 32 (2at
− ag2 − ag−2 ) +
at · ag3 =
1 23 (at 13 28 ,
7 25 wρ ,
+ ag−1 + at + ag + ag2 ),
at , ag , ag−1 , ag2 ,
33 5 2 211 uρ ,
5 2 25 uρ ,
+ ag3 − aρ3 ), af 3 · uρ2 = 0, (at , ag ) =
(at , ag3 ) =
1 23 ,
Table 1: Norton-Sakuma algebras.
7
1 23
+ 3ag − ag2 − ag−1 − ag−2 ) + wρ ,
wρ · wρ =
(at , ag2 ) =
1 25 ,
(aρ3 , uρ2 ) = 0
5 28 ,
The scaling of the products of Table 1 coincides with the one used in [IPSS10]. The missing products of basis vectors may be obtained using the symmetries of the algebras and their mutual inclusions: 2A ,→ 4B, 2B ,→ 4A, 2A ,→ 6A, 3A ,→ 6A. For the rest of the paper, denote by VN X the Norton-Sakuma algebra of type N X. Denote by idN X the identity of VN X . These identities are given explicitly in Table 9 in Appendix A. Denote the automorphism group of VN X by Aut(N X). The following result is Theorem 4.1 in [CR13]. Proposition 3.2. For 2 ≤ N ≤ 6 and X ∈ {A, B, C}, let (D, T, VN X , ϕ, ψ) be the Majorana representation of the dihedral group of order 2N , where VN X = hhat , ag ii, t, g ∈ T , is a Norton-Sakuma algebra of type N X. The following statements hold: 1. The algebra V2A has exactly 8 idempotents and Aut (2A) = Sym {at , ag , atg } ∼ = S3 . 2. The algebra V3A has exactly 16 idempotents and Aut (3A) = ϕ (D) ∼ = S3 . 3. The algebra V3C has exactly 8 idempotents and Aut (3C) = ϕ (D) ∼ = S3 . 4. The algebra V4A has an infinite family of idempotents plus 18 extra idempotents. In this case,
with φ4A
Aut (4A) = hϕ (t) , φ4A i ∼ = D8 , = (at , ag ) ag−1 , ag2 .
5. The algebra V4B has exactly 32 idempotents and
with φ4B
Aut (4B) = hϕ (t) , φ4B i ∼ = D8 , = (at , ag ) ag−1 , ag2 .
6. The algebra V5A has exactly 44 idempotents and
with φ5A order 20.
Aut (5A) = hϕ (t) , ϕ (g) , φ5A i , = ag , ag2 , ag−1 , ag−2 , is isomorphic to the Frobenius group of
7. The algebra V6A has exactly 208 idempotents and
with φ6A
Aut (6A) = hϕ (t) , φ6A i ∼ = D12 = (at , ag ) ag−1 , ag2 ag−2 , ag3 . 8
The next result was verified by direct computations in [MAP], using the idempotents found in [CR13]. Lemma 3.3. Every idempotent of every Norton-Sakuma algebra is semisimple. In view of Lemma 3.3, the spectra of the idempotents of VN X gives essential information about the associative subalgebras of VN X . Although we only require the multiplicities of the eigenvalues 0 and 1, we believe that the full spectrum of each idempotent may be of general interest. In the rest of the paper, we give the spectra of the idempotents of VN X as multisets and, because of Lemma 2.4 (ii), we organize them in terms of Aut(N X)-orbits. Furthermore, we only consider half of the non-trivial nonidentity idempotents; the spectra of the missing idempotents may be found using Lemma 2.4 (iii). Lemma 3.4. For N X ∈ {2A, 3A, 3C}, the following statements hold: (i) The identity of VN X is the unique decomposable idempotent of VN X (ii) There are no non-trivial associative subalgebras of VN X . 1 while the Majorana Proof. The Majorana axes in V have spectrum 0, 1, 2A 4 1 axes in V3C have spectrum 0, 1, 32 . Table 2 gives the spectra of half of the non-trivial non-identity idempotents of V3A , where y3A is defined in Appendix A. Therefore, the result follows by Proposition 2.5 and Corollary 2.8. Orbit
Size
Length
Spectrum
[at ]
3
1
[uρ ]
1
8 5
[y3A ]
3
8 5
1 0, 1, 14 , 32 0, 1, 13 , 31 13 0, 1, 13 , 16
Table 2: Spectra of the idempotents of V3A .
3.1
The Norton-Sakuma Algebra of Type 4A
The Norton-Sakuma algebra V = V4A has an infinite family of idempotents of (1) length 2. In particular, for any λ ∈ [− 53 , 1], we have an idempotent y4A (λ) ∈ (1) V4A , as defined in Appendix A. The spectrum of y4A (λ) is {0, 1, where h(λ) :=
1 , h (λ) , h (λ)} 2
p 1 (17 − 5λ − 5 −15λ2 + 6λ + 9). 5 2
Lemma 3.5. The following statements hold: (i) For any λ ∈ [− 35 , 1], λ 6= 0, 32 , the algebra Vy(1) (λ) is maximal associative. 4A
9
(1)
(ii) The idempotent y := y4A
2 5
is indecomposable with d (y) = 2.
Proof. The equation h(λ) = 1 has no solutions while h (λ) = 1 has the unique solution λ = 0. On the other hand, the equation h(λ) = 0 has the unique solution λ = 52 while h (λ) = 0 has no solutions. Therefore, 0 and 1 are simple (1) eigenvalues of y4A (λ), for any λ ∈ [− 35 , 1], λ 6= 0, 23 . The result follows by Corollary 2.8. The previous lemma implies that, for every idempotent x ∈ V4A , the algebra (x) V0 has finitely many idempotents. The following identity follows from the definition of y: y = id4A − at − ag2 . Table 3 sumarises the spectra of the non-zero non-identity idempotents of (2) V4A , where y4A is defined in Appendix A. Orbit
Size
Length
[at ]
4
1
(1)
2
2
(2)
4
12 7
[y4A (λ)] [y4A ]
Spectrum 1 0, 0, 1, 41 , 32 {0, 1, h(λ), h (λ)} 1 5 6 0, 1, 14 , 14 ,7
Table 3: Spectra of the idempotents of V4A . Since d(x) ≤ 2 for every idempotent x ∈ V4A , Lemma 2.9 implies that every associative subalgebra of V4A is at most three-dimensional. Lemma 3.6. The subalgebras of V4A , hhat , ag2 , id4A − at − ag2 ii and
ag , ag−1 , id4A − ag − ag−1
,
are maximal associative. Proof. The idempotents generating any of the above subalgebras are clearly pairwise orthogonal and their sum is id4A . Moreover, they are indecomposable by Table 3 and Proposition 2.5. The result follows by Theorem 2.7. Lemma 3.7. The Norton-Sakuma algebra of type 4A has infinitely many maximal associative subalgebras. However, it has only two non-trivial maximal associative subalgebras. Proof. The first part of this lemma follows by Lemma 3.5. By Table 3 and Lemma 3.3, only the idempotents in the orbits [at ] and [id4A − at − ag2 ] have a two-dimensional 0-eigenspace, so they are the only idempotents with nonmaximal trivial subalgebra. By Lemma 2.10, each one of these idempotents is contained in the orthogonal basis of at most one three-dimensional maximal associative subalgebra of V4A . Lemma 3.6 describes such algebras, so the result follows.
10
3.2
The Norton-Sakuma Algebra of Type 4B
The Norton-Sakuma algebra V = V4B contains a subalgebra of type 2A with basis {at , ag2 , aρ2 }. Let id2A be the identity of this subalgebra. Table 4 gives the spectra of half of the non-zero non-identity idempotents of V4B , where y4B is defined in Appendix A. Orbit
Size
Length
[at ] aρ2
4
1
1
1
[id2A ]
2
12 5
[id2A − at ]
4
7 5
[y4B ]
4
21 11
Spectrum 1 0, 0, 1, 14 , 32 0, 0, 1, 14 , 14 0, 1, 1, 1, 41 7 0, 0, 1, 34 , 32 1 21 9 0, 1, 11 , 22 , 22
Table 4: Spectra of the idempotents of V4B . Lemma 3.8. Let φ4B ∈ Aut(4B) be the automorphism defined in Proposition 3.2. The subalgebras of V4B , (1)
4B U4B := hhat , id2A − at , idφ2A − aρ2 ii,
(2)
4B U4B := hhaρ2 , id2A − aρ2 , idφ2A − aρ2 ii,
are maximal associative. Proof. Since φ4B 4B 4B (id2A , id2A ) = (aρ2 , idφ2A ) = 1 and (at , idφ2A )=
1 , 8
we have that 4B 4B (at , id2A − at ) = (at , idφ2A − aρ2 ) = (id2A − at , idφ2A − aρ2 ) = 0,
and 4B 4B (aρ2 , id2A − aρ2 ) = (aρ2 , idφ2A − aρ2 ) = (id2A − aρ2 , idφ2A − aρ2 ) = 0.
As the relation 4B id4B = id2A + idφ2A − aρ2
holds, we have that 4B id4B = at + (id2A − at ) + (idφ2A − aρ2 ), 4B id4B = aρ2 + (id2A − aρ2 ) + (idφ2A − aρ2 ).
(1)
(2)
By Table 4 and Proposition 2.5, the idempotents generating U4B and U4B are indecomposable, so the result follows by Theorem 2.7.
11
Lemma 3.9. Every associative subalgebra of V4B is at most three-dimensional. Proof. Suppose U is an associative subalgebra of V6A of dimension k ≥ 4. Let {xi : 1 ≤ i ≤ k} be the orthogonal basis of idempotents of U . Without loss of generality, we may assume U is maximal associative. By Theorem 2.7, id4B ∈ U k P and xi = id4B . By Lemma 2.2 we have that i=1 k X
l(xi ) = l(id4B ) =
i=1
19 . 5
(1)
The orthogonal basis of U contains at most one idempotent of length 1, since there is no pair of orthogonal idempotents of length 1 in V4B . The non-zero idempotents with the smallest length different from 1 are [id2A −at ]∪[id2A −aρ2 ] and the all have length 57 . Therefore, k X
l(xi ) ≥ 1 + 3 ·
i=1
26 19 7 = > , 5 5 5
which contradicts (1). If U is a subalgebra of any Majorana representation V , we introduce the notation [U ] := {U g : g ∈ Aut(V )}. Lemma 3.10. The Norton-Sakuma algebra of type 4B has exactly 9 maximal associative subalgebras; 4 of these subalgebras are trivial associative while 5 are three-dimensional. Proof. The 4 trivial maximal associative subalgebras are the ones in the orbit (1) (2) (1) [Vy4B ]. If U4B and U4B are the subalgebras of Lemma 3.8, the orbits [U4B ] and (2) [U4B ] contain 4 and 1 maximal associative subalgebras respectively. We will show that there are no more maximal associative subalgebras. By Lemma 3.9, there are no four-dimensional associative subalgebras of V4B . Let Nx be the (1) (2) number of associative algebras in [U4B ] ∪ [U4B ] where the idempotent x ∈ V is contained. The following table gives the values of d(x) and Nx for the orbit representatives of idempotents of V4B with d(x) ≥ 2: Idempotent x
d(x)
Nx
Idempotent x
d(x)
Nx
at
2
1
id2A − aρ2
3
3
aρ2
2
1
id2A − at
2
1
If Mx is the number of three-dimensional maximal associative subalgebras of V4B where the idempotent x is contained, Lemma 2.10 shows that Nx ≤ Mx ≤ 2d(x)−1 − 1, whenever dx ≥ 2. However, the above table shows that Nx = 2d(x)−1 − 1, so Nx = Mx . The result follows. 12
3.3
The Norton-Sakuma Algebra of Type 5A
Table 5 gives the spectra of half of the non-zero non-identity idempotents of (i) V5A , where y5A , for i = 1, 2, 3, are defined in Appendix A. Orbit
Size
Length
[at ]
5
1
(1)
2
16 7
(2)
10
25 14
(3)
10
16 7
[y5A ] [y5A ] [y5A ]
Spectrum 1 1 0, 0, 1, 14 , 32 , 32 0, 1, 53 , 35 , 25 , 25 5 57 3 , 64 , 8 0, 0, 1, 64 0, 1, 87 , 35 , 25 , 18
Table 5: Spectra of the idempotents of V5A . Lemma 3.11. The subalgebra of V5A , (2)
(2)
hhat , y5A , (y5A )φ5A ii, is maximal associative. Proof. The result follows by Table 5, Proposition 2.5 and Theorem 2.7, since (2)
(2)
(2)
(2)
(at , y5A ) = (at , (y5A )φ5A ) = (y5A , (y5A )φ5A ) = 0 and (2)
(2)
id5A = at + y5A + (y5A )φ5A .
Proposition 3.12. The Norton-Sakuma algebra of type 5A has exactly 11 maximal associative subalgebras; 6 of these algebras are trivial while 5 are threedimensional. Proof. By Table 5 and Corollary 2.8, the trivial maximal associative subalgebras of V5A are exactly the ones in the orbits [Vy(i) ] for i = 1 and i = 3, which have 5A sizes 1 and 5 respectively. Using the action of Aut(5A), Lemma 3.11 defines 5 three-dimensional maximal associative subalgebras of V5A . The result follows using Lemma 2.9 and Lemma 2.10.
3.4
The Norton-Sakuma Algebra of Type 6A
The spectra of half of the non-trivial non-zero idempotents of V6A is given in (i) Table 6. The idempotents y6A , for 1 ≤ i ≤ 8, are given in Appendix A. In this case, id2A and id3A are the identities of the subalgebras of types 2A and 3A with bases {at , ag3 , aρ3 } and {at , ag2 , ag−2 , uρ2 } respectively.
13
Orbit
Size
Length
Spectrum
[at ]
6
1
1 1 {0, 0, 0, 1, 41 , 14 , 32 , 32 }
[aρ3 ]
1
1
{0, 0, 0, 0, 1, 14 , 14 , 14 }
[uρ2 ]
1
8 5
{0, 0, 0, 1, 31 , 13 , 13 , 13 }
[aρ3 + uρ2 ]
1
13 5
7 7 {0, 1, 1, 14 , 13 , 13 , 12 , 12 }
[id2A ]
3
12 5
3 3 1 {0, 1, 1, 1, 41 , 10 , 10 , 20 }
[id2A − at ]
6
7 5
3 1 7 3 {0, 0, 1, 34 , 10 , 20 , 32 , 160 }
[id2A − aρ3 ]
3
7 5
3 1 1 {0, 0, 0, 1, 43 , 10 , 20 , 20 }
[id3A ]
2
116 35
5 5 5 {0, 1, 1, 1, 1, 14 , 14 , 14 }
[id3A − at ]
6
81 35
5 3 31 73 {0, 0, 1, 34 , 14 , 28 , 32 , 224 }
[id3A − uρ2 ]
2
12 7
1 1 5 {0, 0, 1, 23 , 23 , 42 , 42 , 14 }
[y3A ]
6
8 5
1 13 {0, 0, 0, 1, 31 , 13 , 16 , 16 }
[id3A − y3A ]
6
12 7
3 5 1 33 {0, 0, 1, 23 , 16 , 14 , 42 , 112 }
(1)
3
116 35
23 23 5 5 {0, 1, 1, 1, 28 , 28 , 14 , 14 }
(1)
3
81 35
5 3 3 23 {0, 0, 1, 47 , 14 , 4 , 28 , 28 }
(2)
6
13 5
3 27 7 {0, 1, 1, 14 , 13 , 32 , 32 , 12 }
(3)
6
12 7
5 1 1 85 {0, 0, 1, 23 , 14 , 42 , 112 , 112 }
(4)
6
11 6
1 1 11 7 7 {0, 0, 1, 12 , 12 , 12 , 18 , 18 }
(5)
6
97 30
2 29 7 {0, 1, 1, 56 , 31 45 , 15 , 30 , 18 }
(6)
6
21 11
3 1 13 1 7 35 {0, 1, 11 , 22 , 22 , 44 , 44 , 44 }
(7)
12
≈ 2.1742
{0, 1, λi : 1 ≤ i ≤ 6}
(8)
12
≈ 2.6787
{0, 1, µi : 1 ≤ i ≤ 6}
[y6A ] [y6A − aρ2 ] [y6A ] [y6A ] [y6A ] [y6A ] [y6A ] [y6A ] [y6A ]
Table 6: Spectra of the idempotents of V6A . Let Br (c) denote the interval centered at c ∈ R with radius r ∈ R. With (7) (7) respect to the eigenvalues of y6A and y6A , we have that, for r = 10−7 , λ1 λ2 λ3 λ4 λ5 λ6
∈ Br (0.940315847805478), ∈ Br (0.717053658733658), ∈ Br (0.381819823054333), ∈ Br (0.260998283012905), ∈ Br (0.0207051872307907), ∈ Br (0.0656436413846592),
µ1 µ2 µ3 µ4 µ5 µ6 14
∈ Br (0.0384530272052537), ∈ Br (0.236296198501829), ∈ Br (0.368766795619826), ∈ Br (0.731570501304047), ∈ Br (0.955825154859771), ∈ Br (0.991047562739161).
Lemma 3.13. Every associative subalgebra of V6A is at most 3-dimensional. Proof. The result follows by a similar argument as the one used in Lemma 3.9, since 57 is the smallest length greater that 1 of an idempotent in V6A and l(id6A ) = 51 10 . Lemma 3.14. The subalgebras of V6A given in Table 7 are maximal associative. Proof. This follows by Table 6, Proposition 2.5 and Theorem 2.7.
Associative subalgebra
Orbit size
hhaρ3 , uρ2 , id6A − aρ3 − uρ2 ii
1
hhaρ3 , id6A − id2A , id2A − aρ3 ii
3
(1)
(1)
hhaρ3 , y6A − aρ3 , id6A − y6A ii
3
hhuρ2 , id3A − uρ2 , id6A − id3A ii
2
hhat , id2A − at , id6A − id2A ii
6
hhat , id3A − at , id6A − id3A ii
6
hhat , id6A − y6A , (y3A )φ6A ii
(2)
6
hhy3A , id3A − y3A , id6A − id3A ii
6
(1)
(3)
hhy6A , (y3A )φ6A , id6A − y6A ii (4)
(5)
ϕ(g)
hhy6A , id6A − y6A , id2A − aρ3 ii
6 6
Table 7: Non-trivial maximal associative subalgebras of V6A . Lemma 3.15. The Norton-Sakuma algebra of type 6A has exactly 75 maximal associative subalgebras; 30 of these algebras are trivial while 45 are threedimensional. Proof. By Table 6 and Corollary 2.8, the trivial maximal associative subalgebras of V6A are contained in the orbits [Vy(i) ] for i = 6, 7, 8, of sizes 6, 12 and 6A 12, respectively. Using the action of Aut(6A), Table 7 defines 45 non-trivial maximal associative subalgebras of V6A . By Lemma 3.13, there are no fourdimensional associative subalgebras of V6A . Now we show that there are no more three-dimensional associative subalgebras in V6A . For each idempotent x ∈ V6A , let Nx be the number of three-dimensional associative subalgebras defined by Table 7 where x is contained. Table 8 contains the values d(x) and Nx for the orbit representatives of idempotents of V6A with d(x) ≥ 2. As Nx = 2d(x)−1 − 1 for every idempotent x ∈ V with d(x) ≥ 2, Lemma 2.10 implies that there are no more three-dimensional associative subalgebras of V6A besides the algebras of Table 7.
15
Idempotent x
d(x)
Nx
Idempotent x
d(x)
Nx
at
3
3
id3A − uρ2
2
1
aρ3
4
7
y3A
3
3
uρ2
3
3
id3A − y3A
2
1
1
id6A −
(1) y6A
3
3
3
(1) y6A
2
1
2
1
2
1
2
1
2
1
id6A − aρ3 − uρ2 id6A − id2A id2A − at
2 3 2
1
id6A − (3) y6A (4) y6A
id6A − id2A + aρ3
3
3
id6A − id3A
4
7
id3A − at
2
− aρ2 (2) y6A
id(6A) −
1
(5) y6A
Table 8: Values of d(x) and Nx for idempotents x ∈ V6A .
4
Conclusions
The following theorem sumarises the main results of this paper: Theorem 4.1. Consider the Norton-Sakuma algebras VN X . The following statements hold: (i) Every idempotent of every Norton-Sakuma algebra is semisimple. (ii) There are no four-dimensional associative subalgebras in any of the NortonSakuma algebras. (iii) There are no non-trivial associative subalgebras of V2A , V3A and V3C . (iv) There are exactly 2 non-trivial maximal associative subalgebras of V4A . (v) There are exactly 5 non-trivial maximal associative subalgebras of V4B . (vi) There are exactly 5 non-trivial maximal associative subalgebras of V5A . (vii) There are exactly 45 non-trivial maximal associative subalgebras of V6A .
16
A
Appendix
This appendix describes explicitly the idempotents of the Norton-Sakuma algebras that were calculated in [CR13]. Table 9 gives the identities of the Norton-Sakuma algebras. Identity
Length
id2A := 54 (at + ag + aρ )
12 5
id2B := at + ag
2
id3A :=
16 21 (at
+ ag + ag−1 ) +
id3C :=
32 33 (at
+ ag + ag−1 )
2
3 14 uρ
116 35 35 11
id4A := 45 (at + ag + ag−1 + ag2 ) + 25 vρ
4
id4B := 45 (at + ag + ag−1 + ag2 ) + 35 aρ2
19 5
id5A :=
32 35 (at
+ ag + ag−1 + ag2 + ag−2 )
id6A := 32 (at + ag + ag−1 + ag2 + ag−2 + ag3 ) + 21 aρ3 + 83 uρ2
32 7 51 10
Table 9: Identities of the Norton-Sakuma algebras. The idempotent y3A ∈ V3A is given by y3A :=
2 1 (4at + 4ag + ag−1 ) − uρ . 9 4
For any λ ∈ [− 35 , 1], the algebra V4A contains an idempotent (1)
y4A (λ) := f (λ)(at + ag2 ) + f (λ)(ag + ag−1 ) + λvρ where
1 1p (1 − λ) − −15λ2 + 6λ + 9, 2 6 √ and f (λ) is the conjugate of f (λ) in Q −15λ2 + 6λ + 9 . Furthermore, √ √ 2 2 2 (2) y4A := 2 − 2 (at + ag ) + 2 + 2 ag−1 + ag2 − vρ , 7 7 7 is an idempotent of V4A . The following is an idempotent of V4B √ √ 4 4 5 y4B := 1 + 2 (at + ag ) + 1 − 2 ag−1 + ag2 + aρ2 . 11 11 11 The following elements are idempotents of V5A : f (λ) :=
(1)
211 √ 16 at + ag + ag−1 + ag2 + ag−2 + 5wρ , 35 175 √ 1 := −3at + 2α ag + ag−1 + 2α ag2 + ag−2 − 256 5wρ , 70 √ 4 := 20at + β ag + ag−1 + β ag2 + ag−2 − 384 5wρ , 175
y5A := (2)
y5A
(3)
y5A
17
√ √ √ where α1 = 16+7 5, β1 = 20−7 5, and α1 , β1 are their conjugates in Q 5 . The following are idempotents of V6A : 1 (16(ag + ag−1 + ag2 + ag−2 ) + 4(at + ag3 ) + 12aρ3 − 9uρ2 ), 21 1 (2) y6A := (36at + 32(ag + ag−1 ) + 8ag3 − 9uρ2 ), 36 1 (3) y6A := (48(at + 4ag2 + 4ag−2 ) − 8(ag3 + 4ag + 4ag−1 ) + 144aρ3 − 45uρ2 ), 252 1 (4) y6A := (16[α(at + ag ) + α(ag−2 + ag3 ) − (ag−1 + ag2 )] + 36aρ3 + 45uρ2 ), 216 1 (5) y6A := (80[α(at + ag ) + α(ag−2 + ag3 )] + 784(ag−1 + ag2 ) − 36aρ3 + 225uρ2 ), 1080 1 (6) y6A := (16[β(at + ag ) + β(ag−2 + ag3 )] − 8(ag−1 + ag2 ) + 6aρ3 + 45uρ2 ), 66 √ √ √ where α = 5 + 22 3, β = 1 + 3, and α, β are their conjugates in Q 3 . Moreover, it was shown in [CR13] that there is an idempotent (1)
y6A
:=
(7)
y6A := γ1 at + γ2 ag + γ3 ag−1 + γ4 ag2 + γ5 ag−2 + γ6 ag3 + γ7 aρ3 + γ8 aρ2 , with γ1 γ3 γ5 γ7
∈ Br (0.118600343195), ∈ Br (0.672945208716), ∈ Br (0.034809133018), ∈ Br (−0.258738375363),
γ2 γ4 γ6 γ8
∈ Br (0.116899056660), ∈ Br (0.891963849266), ∈ Br (0.960846592395), ∈ Br (−0.226937866453),
for r = 10−10 , and an idempotent (8)
y6A := δ1 at + δ2 ag + δ3 ag−1 + δ4 ag2 + δ5 ag−2 + δ6 ag3 + δ7 aρ3 + δ8 aρ2 , with δ1 δ3 δ5 δ7
∈ Br (0.753 376146443), ∈ Br (−0.153112021089), ∈ Br (0.110690245253), ∈ Br (0.620 071135272),
δ2 δ4 δ6 δ8
18
∈ Br (−0.031896831434), ∈ Br (0.729547069626), ∈ Br (0.844782757936), ∈ Br (−0.121749860276).
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