A note on 2-generated symmetric axial algebras of Monster type
aa r X i v : . [ m a t h . R A ] J a n A NOTE ON -GENERATED SYMMETRIC AXIAL ALGEBRASOF MONSTER TYPE CLARA FRANCHI AND MARIO MAINARDIS
Abstract.
In [10], Yabe gives an almost complete classification of primitivesymmetric 2-generated axial algebras of Monster type. In this note, we con-struct a new infinite-dimensional primitive 2-generated symmetric axial alge-bra of Monster type (2 , ) over a field of characteristic 5, and use this algebrato complete the last case left open in Yabe’s classification. Introduction
Axial algebras of Monster type have been introduced in [4] by Hall, Rehren andShpectorov, in order to generalise some subalgebras of the Griess algebra (definedMajorana algebras by Ivanov [5]) and create a new tool for better understanding,and possibly unifying, the classification of finite simple groups. They have recentlyappeared also in other branches of mathematics (see [8, 9]). In [6, 7] Rehren starteda systematic study of primitive 2-generated axial algebras of Monster type, con-structing several classes of new algebras. More examples have been found by Galtet al. [3], and, independently, by Yabe [10]. In [10], Takahiro Yabe obtained an al-most complete classification of the primitive 2-generated symmetric axial algebrasof Monster type. Yabe left open only the case of algebras over a field of character-istic 5 and axial dimension greater than 5. Indeed something surprising happensin the latter case, namely we show that, over a field F of characteristic 5, thereexists a new infinite-dimensional primitive 2-generated symmetric axial algebra ˆ H of Monster type (2 , ) such that any primitive 2-generated symmetric axial alge-bras of Monster type (2 , ) is isomorphic to a quotient of ˆ H (in particular, theHighwater algebra H [1] is a proper factor of ˆ H over an infinite-dimensional ideal).As a corollary we complete Yabe’s classification.For the definitions and further motivation refer to [4, 6, 7], for the notation andthe basic properties of axial algebras refer to [1]. In particular, throughout thispaper, F is a field of characteristic 5. For a 2-generated symmetric axial algebra V of Monster type ( α, β ) over F , let a , a be the generating axes of V . For i ∈ { , } ,let τ i be the Miyamoto involution associated to a i . Set ρ := τ τ , and, for i ∈ Z , a i := a ρ i and a i +1 := a ρ i . Note that, since ρ is an automorphism of V , for every j ∈ Z , a j is an axis. Denote by τ j := τ a j the corresponding Miyamoto involution.The algebra V is symmetric if it admits an algebra automorphism f that swaps a and a , whence, for every i ∈ Z , a fi = a − i +1 . Let σ i be the element of h τ , f i that swaps a with a i . Since, by [1, Lemma 4.2], for n ∈ Z + and i, j ∈ Z such that i ≡ n j , a i a i + n − β ( a i + a i + n ) = a j a j + n − β ( a j + a j + n ) , we can define(1) s ¯ ı,n := a i a i + n − β ( a i + a i + n ) . where ¯ ı denotes the congruence class i + n Z . Since V is primitive, there is alinear function λ a : V → F such that every v can be written in a unique way as v = λ a ( v ) a + v + v α + v β , where v , v α , w β are 0-, α -, β -eigenvectors for ad a ,respectively. For i ∈ Z , set λ i := λ a ( a i ) and let(2) a i = λ a ( a i ) a + u i + v i + w i be the decomposition of a i into ad a -eigenvectors, where u i is a 0-eigenvector, v i isan α -eigenvector and w i is a β -eigenvector.Following Yabe [10], we denote by D the positive integer such that { a , . . . , a D − } is a basis for the linear span h a i | i ∈ Z i of the set of the axes a i ’s. D is called the axial dimension of V . Our results are the following Theorem 1.
Let V be a primitive -generated symmetric axial algebra of Monstertype (2 , ) over a field of characteristic . If λ = λ = 1 , then V is isomorphic toa quotient of the algebra ˆ H . Corollary 2.
Let V be a primitive -generated symmetric axial algebra of Monstertype ( α, β ) over a field of characteristic . If D ≥ , then V is isomorphic to aquotient of one of the following:(1) the algebra A α , as defined in [7] ;(2) the algebra V ( α ) , as defined in [3] ;(3) the algebra ˆ H , as defined in Section 2. Note that, for every α , Rehren’s algebra 6 A α and Yabe’s algebra VI ( α, − α α − )coincide and the 8-dimensional algebra V ( α ) of type ( α, α ) constructed in [3] co-incides with Yabe’s algebra VI ( α, α ). Furthermore, remarkably, over a field ofcharacteristic 5, the Highwater algebra H (see [2] and [10]) is isomorphic to a quo-tient of ˆ H , Yabe’s algebras V (2 , ) and V (2 , ), and Rehren’s algebra 5 A areall isomorphic, and are in turn a quotient of H . Finally, also the algebra 6 A is aquotient algebra of ˆ H . 2. The algebra ˆ H In this section, for every i ∈ Z , denote by ¯ ı the congruence class i + 3 Z . Let ˆ H be an infinite-dimensional F -vector space with basis B := { ˆ a i , ˆ s ¯0 ,j , ˆ s ¯1 , k , ˆ s ¯2 , k | i ∈ Z , j, k ∈ Z + } , ˆ H := M i ∈ Z F ˆ a i ⊕ M j ∈ Z + F ˆ s ¯0 ,j ⊕ M k ∈ Z + ( F ˆ s ¯1 , k ⊕ F ˆ s ¯2 , k ) . Set ˆ s ¯0 , := 0 and, if j
0, ˆ s ¯1 ,j := ˆ s ¯0 ,j =: ˆ s ¯2 ,j . Let ˆ τ and ˆ f be the linear maps ofˆ H defined on the basis elements byˆ a ˆ τ i = ˆ a − i , (ˆ s ¯0 ,j ) ˆ τ = ˆ s ¯0 ,j , (ˆ s ¯1 , k ) ˆ τ = ˆ s ¯2 , k , and (ˆ s ¯2 , k ) ˆ τ = ˆ s ¯1 , k , ˆ a ˆ fi = ˆ a − i +1 , (ˆ s ¯0 ,j ) ˆ f = ˆ s ¯0 ,j if j , and (ˆ s ¯0 , k ) ˆ f = ˆ s ¯1 , k , (ˆ s ¯1 , k ) ˆ f = ˆ s ¯0 , k , (ˆ s ¯2 , k ) ˆ f = ˆ s ¯2 , k . -GENERATED AXIAL ALGEBRAS 3 Define a commutative non-associative product on ˆ H extending by linearity thefollowing values on the basis elements (where δ ¯ ı ¯ r denotes the Kronecker delta and( − ∗ ¯1 := −
1, ( − ∗ ¯2 := 1, and 0 ∗ ¯ t := 0 for every t ∈ Z ):( ˆ H ) ˆ a i ˆ a j := − a i + ˆ a j ) + ˆ s ¯¯ ı, | i − j | ,( ˆ H ) ˆ a i ˆ s ¯ r,j := − a i + (ˆ a i − j + ˆ a i + j ) − ˆ s ¯ r,j − ( δ ¯ ı ¯ r − ∗ (¯ ı − ¯ r )(ˆ s ¯ r − ¯1 ,j − ˆ s ¯ r +¯1 ,j ),( ˆ H ) ˆ s ¯ r,i ˆ s ¯ t,j := 2(ˆ s ¯ r,i + ˆ s ¯ t,j ) − s ¯0 , | i − j | + ˆ s ¯1 , | i − j | + ˆ s ¯2 , | i − j | + ˆ s ¯0 ,i + j + ˆ s ¯1 ,i + j + ˆ s ¯2 ,i + j ),if { i, j } 6⊆ Z ,( ˆ H ) ˆ s ¯0 , h ˆ s ¯0 , k := 2(ˆ s ¯0 , h + ˆ s ¯0 , k ) − (ˆ s ¯0 , | h − k | + ˆ s ¯0 , h + k ) ),( ˆ H ) ˆ s ¯0 , h ˆ s ¯1 , k := 2(ˆ s ¯0 , h + ˆ s ¯1 , h − ˆ s ¯2 , h + ˆ s ¯0 , k + ˆ s ¯1 , k − ˆ s ¯2 , k ) − (ˆ s ¯0 , | h − k | +ˆ s ¯1 , | h − k | − ˆ s ¯2 , | h − k | + ˆ s ¯0 , h + k ) + ˆ s ¯1 , h + k ) − ˆ s ¯2 , h + k ) ),( ˆ H ) ˆ s ¯0 , h ˆ s ¯2 , k := 2(ˆ s ¯0 , h − ˆ s ¯1 , h + ˆ s ¯2 , h + ˆ s ¯0 , k − ˆ s ¯1 , k + ˆ s ¯2 , k ) − (ˆ s ¯0 , | h − k | − ˆ s ¯1 , | h − k | + ˆ s ¯2 , | h − k | + ˆ s ¯0 , h + k ) − ˆ s ¯1 , h + k ) + ˆ s ¯2 , h + k ) ),( ˆ H ) ˆ s ¯1 , h ˆ s ¯2 , k := 2( − ˆ s ¯0 , h + ˆ s ¯1 , h + ˆ s ¯2 , h − ˆ s ¯0 , k + ˆ s ¯1 , k + ˆ s ¯2 , k ) − ( − ˆ s ¯0 , | h − k | +ˆ s ¯1 , | h − k | + ˆ s ¯2 , | h − k | − ˆ s ¯0 , h + k ) + ˆ s ¯1 , h + k ) + ˆ s ¯2 , h + k ) ).We now introduce some eigenvectors for ad ˆ a and study how they multiply. For i ∈ Z + , set ˆ u i := − a + (ˆ a i + ˆ a − i ) + 2ˆ s ¯0 ,i , ˆ v i := − a + (ˆ a i + ˆ a − i ) − ˆ s ¯0 ,i , ˆ w i := ˆ a i − ˆ a − i ,u i := − a + (ˆ a − i + ˆ a i ) − (ˆ s ¯0 ,i + ˆ s ¯1 ,i + ˆ s ¯2 ,i ) ,w i := ˆ s ¯1 ,i − ˆ s ¯2 ,i . Then, the ˆ u i ’s and u i ’s are 0-eigenvectors for ad ˆ a , the ˆ v i ’s are 2-eigenvectors forad ˆ a , the ˆ w i ’s and w i ’s are − ˆ a . Note that, if i u i = ˆ u i and w i = 0. Moreover, it will be convenient to use the following notation: for i, j ∈ Z + , set ˆ c j := − a + (ˆ a − j + ˆ a j ) , ˆ c i,j := − c i − c j + ˆ c | i − j | + ˆ c i + j , ˆ σ i,j := ˆ s ¯0 ,i + ˆ s ¯0 ,j + ˆ s − ¯ ı, | i − j | + ˆ s ¯ ı, | i − j | + ˆ s − ¯ ı,i + j + ˆ s ¯ ı,i + j , ˆ u i,j := − u i − u j + ˆ u | i − j | + ˆ u i + j , ˆ v i,j := − v i − v j + ˆ v | i − j | + ˆ v i + j . We collect in the following lemma the main relations among the above vectors.
Lemma 3.
For all i, j ∈ Z + , we have(1) ˆ u j = ˆ c j + 2ˆ s ¯0 ,j ,(2) ˆ v j = ˆ c j − ˆ s ¯0 ,j ,(3) u j = ˆ c j − (ˆ s ¯0 ,j + ˆ s ¯1 ,j + ˆ s ¯2 ,j ) ,(4) ˆ u i,j = ˆ c i,j + ˆ s ¯0 ,i + ˆ s ¯0 ,j + 2ˆ s ¯0 , | i − j | + 2ˆ s ¯0 ,i + j ,(5) ˆ v i,j = ˆ c i,j + 2(ˆ s ¯0 ,i + ˆ s ¯0 ,j ) − (ˆ s ¯0 , | i − j | + ˆ s ¯0 ,i + j ) .(6) ˆ c i ˆ c j = ˆ σ i,j ,(7) ˆ c i ˆ s ¯ r,j = ˆ c i,j = ˆ c j ˆ s ¯ r,i . CLARA FRANCHI AND MARIO MAINARDIS
Proof.
The first five assertions are immediate. A straightforward computation givesthe sixth:ˆ c i ˆ c j = ( − a + (ˆ a − i + ˆ a i ))( − a + (ˆ a − j + ˆ a j ))= − ˆ a − − a − i + ˆ a + ˆ s ¯0 ,i − a i − a + ˆ s ¯0 ,i ) − − a − a − j + ˆ s ¯0 ,j − a − a j + ˆ s ¯0 ,j )+( − a − i − a − j + ˆ s − ¯ ı, | i − j | − a i − a − j + ˆ s ¯ ı,i + j − a − i − a j + ˆ s − ¯ ı,i + j − a i − a j + ˆ s ¯ ı, | i − j | )= ˆ s ¯0 ,i + ˆ s ¯0 ,j + ˆ s − ¯ ı, | i − j | + ˆ s ¯ ı, | i − j | + ˆ s − ¯ ı,i + j + ˆ s ¯ ı,i + j = ˆ σ i,j . Similarly, for the seventh, we haveˆ c i ˆ s ¯ r,j = − (2ˆ a − (ˆ a − i + ˆ a i ))ˆ s ¯ r,j == − − a + (ˆ a − j + ˆ a j ) − ˆ s ¯ r,j − ( δ ¯0¯ r − ∗ (¯0 − ¯ r )(ˆ s ¯ r − ¯1 ,j − ˆ s ¯ r +¯1 ,j )]+( − a − i + (ˆ a − i − j + ˆ a − i + j ) − ˆ s ¯ r,j − ( δ − ¯ ı ¯ r − ∗ ( − ¯ ı − ¯ r )(ˆ s ¯ r − ¯1 ,j − ˆ s ¯ r +¯1 ,j )+( − a i + (ˆ a i − j + ˆ a i + j ) − ˆ s ¯ r,j − ( δ ¯ ı ¯ r − ∗ (¯ ı − ¯ r )(ˆ s ¯ r − ¯1 ,j − ˆ s ¯ r +¯1 ,j ) . Here, ˆ s ¯ r,j , ˆ s ¯ r − ¯1 ,j , and ˆ s ¯ r +¯1 ,j cancel and the first equality of the last claim followsafter the terms are rearranged. Since, by the definition, ˆ c i,j is symmetric in i and j , we have ˆ c i,j = ˆ c j,i = ˆ c j ˆ s ¯ r,i . (cid:3) Note that also ˆ σ i,j is symmetric in i and j . This evident by Lemma 3.(4), sinceˆ H is commutative. Two more relations will be useful in the sequel. Lemma 4.
For all i, j ∈ Z + , we have(1) ˆ s ¯0 ,i (ˆ s ¯0 ,j + ˆ s ¯1 ,j + ˆ s ¯2 ,j ) = ˆ s ¯0 ,i + ˆ s ¯0 ,j + 2(ˆ s ¯0 , | i − j | + ˆ s ¯0 ,i + j ) , (2) ˆ u i − u i = − s ¯0 ,i + ˆ s ¯1 ,i + ˆ s ¯2 ,i .Proof. A straightforward computation gives the claims. (cid:3)
Now are now ready to compute the products of the vectors ˆ u j , u j , and ˆ v j . Lemma 5.
For all i, j ∈ Z + , we have(1) ˆ u i ˆ u j = − ˆ u i,j − u | i − j | − u | i − j | ) − u i + j − u i + j ) ,(2) u i v j = v i,j ,(3) v i v j = − u i,j − (ˆ u | i − j | − u | i − j | ) − (ˆ u i + j − u i + j ) ,(4) ˆ u i u j = − ˆ u i,j ,(5) ˆ v i u j = ˆ v i,j .Proof. By Lemma 3,ˆ u i ˆ u j = (ˆ c i + 2ˆ s ¯0 ,i )(ˆ c j + 2ˆ s ¯0 ,j )= ˆ c i ˆ c j + 2ˆ c i ˆ s ¯0 ,j + 2ˆ s ¯0 ,i ˆ c j − ˆ s ¯0 ,i ˆ s ¯0 ,j = ˆ σ i,j + 2ˆ c i,j + 2ˆ c i,j − ˆ s ¯0 ,i ˆ s ¯0 ,j = ˆ σ i,j − ˆ c i,j − ˆ s ¯0 ,i ˆ s ¯0 ,j . Assume i ≡
0. Then ˆ σ i,j = ˆ s ¯0 ,i + ˆ s ¯0 ,j + 2(ˆ s ¯0 , | i − j | + ˆ s ¯0 ,i + j ). Moreover,(3) ˆ s ¯0 ,i ˆ s ¯0 ,j = 2(ˆ s ¯0 ,i + ˆ s ¯0 ,j ) − (ˆ s ¯0 , | i − j | + ˆ s ¯0 ,i + j ) . This is immediate if j ≡
0, since in this case ( ˆ H ) holds. If j
0, then | i − j | 6≡ i + j
0. Thus ˆ s ¯0 , | i − j | = ˆ s ¯1 , | i − j | = ˆ s ¯2 , | i − j | , ˆ s ¯0 ,i + j = ˆ s ¯1 ,i + j = ˆ s ¯0 ,i + j and ( ˆ H ) -GENERATED AXIAL ALGEBRAS 5 ⋆ −
21 1 2 −
20 0 2 −
22 2 2 0 , − − − − − , , Table 1.
Fusion law for ˆ H reduces to (3). Hence, by Lemma 3.(4)ˆ u i ˆ u j = − ˆ c i,j + ˆ s ¯0 ,i + ˆ s ¯0 ,j + 2(ˆ s ¯0 , | i − j | + 2ˆ s ¯0 ,i + j ) − s ¯0 ,i + ˆ s ¯0 ,j )+(ˆ s ¯0 , | i − j | + ˆ s ¯0 ,i + j )= − ˆ c i,j − (ˆ s ¯0 ,i + ˆ s ¯0 ,j ) − s ¯0 , | i − j | + ˆ s ¯0 ,i + j ) = − ˆ u i,j . Assume i j
0. Then ˆ σ i,j = ˆ s ¯0 ,i + ˆ s ¯0 ,j + ˆ s ¯1 , | i − j | + ˆ s ¯2 , | i − j | + ˆ s ¯1 ,i + j + ˆ s ¯2 ,i + j ,while the product ˆ s ¯0 ,i ˆ s ¯0 ,j is given by ( ˆ H ). Henceˆ u i ˆ u j = − ˆ c i,j + ˆ s ¯0 ,i + ˆ s ¯0 ,j + ˆ s ¯1 , | i − j | + ˆ s ¯2 , | i − j | + ˆ s ¯1 ,i + j + ˆ s ¯2 ,i + j − s ¯0 ,i + ˆ s ¯0 ,j )+2(ˆ s ¯0 , | i − j | + ˆ s ¯1 , | i − j | + ˆ s ¯2 , | i − j | + ˆ s ¯0 ,i + j + ˆ s ¯1 ,i + j + ˆ s ¯2 ,i + j )= − ˆ u i,j − u | i − j | − u | i − j | ) − u i + j − u i + j ) . This proves the first assertion. The remaining assertions follow with similar com-putations, using Lemma 4. (cid:3)
Theorem 6.
The algebra ˆ H defined above is a primitive -generated symmetricaxial algebra of Monster type (2 , ) over any field of characteristic .Proof. Remind that in characteristic 5, = −
2. It is easy to see that the maps ˆ τ and ˆ f are algebra automorphisms of ˆ H and that the map ˆ θ := τ ˆ f induces on theset { ˆ a i | i ∈ Z } the translation ˆ a i ˆ a i +1 . Let H := hh a , a ii be the subalgebraof ˆ H generated by ˆ a and ˆ a . Note that ˆ s ¯0 , = ˆ a ˆ a + 2(ˆ a + ˆ a ) ∈ H . Also,ˆ a − = ˆ a ˆ s ¯0 , + 2ˆ a − ˆ a + ˆ s ¯0 , ∈ H . This gives us ˆ a − ∈ H . Clearly, H = hh ˆ a , ˆ a ii is invariant under ˆ f and also H = hh ˆ a − , ˆ a , ˆ a ii is invariant under the involutionˆ τ . Thus H is invariant under ˆ θ and so H contains all the ˆ a i ’s. It follows that H contains all the s ¯ r,j , that is H = ˆ H .Since, for every i ∈ Z , ˆ a i = ˆ a ˆ θ i , to show that ˆ H is an axial algebra of Monstertype (2 , ) it is enough to prove that ˆ a is an axis with respect to the Monsterfusion law in Table 1. Further, for every i ∈ Z + , we have(4) h ˆ a , ˆ a i , ˆ a − i , ˆ s ¯0 ,i i = h ˆ a , ˆ u i , ˆ v i , ˆ w i i if i h ˆ a , ˆ a i , ˆ a − i , ˆ s ¯0 ,i , ˆ s ¯1 ,i , ˆ s ¯2 ,i i = h ˆ a , ˆ u i , ˆ v i , ˆ w i , u i , w i i if i ≡ . Hence, a basis of ad ˆ a -eigenvectors for ˆ H is given byˆ a , ˆ u i , ˆ v i , ˆ w i , u k , w k , with i, k ∈ Z + . CLARA FRANCHI AND MARIO MAINARDIS
In particular, since ˆ a is the unique element of this basis that is a 1-eigenvector forad ˆ a , the algebra is primitive. Finally, set H := h ˆ u i , u i | i ∈ Z + i , H := h ˆ v i | i ∈ Z + i , H − := h ˆ w i , w i | i ∈ Z + i . Then, for z ∈ { , , − } , H z is the z -eigenspace for ad ˆ a and ˆ τ acts as the identityon h ˆ a i ⊕ H ⊕ H and as the multiplication by − H − . Since τ is an algebraautomorphism, we have H z H − ⊆ H − for every z ∈ { , } and H − H − ⊆ h ˆ a i ⊕ H ⊕ H . By Lemma 5, we also have H H ⊆ H , H H ⊆ H , and H H ⊆ H .Hence ˆ H respects (a restricted version of) the Monster fusion law and the result isproved. (cid:3) Note that h ˆ s ¯0 , k − ˆ s ¯1 , k , ˆ s ¯0 , k − ˆ s ¯2 , k , ˆ s ¯1 , k − ˆ s ¯2 , k | k ∈ Z + i is an ˆ f -invariantideal of ˆ H and the corresponding factor algebra is isomorphic to the Highwateralgebra H . Moreover, the algebra 6 A is isomorphic to the factor of ˆ H over theideal linearly spanned by the vectorsˆ a i − ˆ a i − , for i ≥ , ˆ s ¯0 , − ˆ s ¯0 , , ˆ s ¯0 , − ˆ s ¯0 , , ˆ s ¯0 ,j − ˆ s ¯0 ,j − , for j ≥ , ˆ x, ˆ x ˆ f , ˆ x ˆ f ˆ τ , where ˆ x := ˆ s ¯0 , − ˆ s ¯0 , − ˆ s ¯0 , + ˆ a − + ˆ a − + ˆ a + ˆ a − a + ˆ a ) . Proofs of the main results
In the next lemma we recall some basic properties of the elements s ¯ r,n that willbe used throughout the proof of Theorem 1 without further reference. Lemma 7.
Let V be a primitive -generated symmetric axial algebra of Monstertype. For every n ∈ Z + and i ∈ Z the following hold(1) ( s ¯ r,n ) σ i = s ¯ k,n , with r + i ≡ k mod n ;(2) the group h τ , f i acts transitively on the set { s ¯ r,n | ¯ r ∈ Z /n Z } , for each n ∈ Z + ;(3) λ a ( s ¯0 ,n ) = λ n − β − βλ n .Proof. The first assertion is Lemma 4.2 in [1] and (2) and (3) follow immediately.For the last one, note that, for every x ∈ V , we have λ a ( a x ) = λ a ( x ) (thisfollows immediately from the linearity of λ a , decomposing x into a sum of ad a -eigenvectors). Hence, by the definition of s ¯0 ,n , we get λ a ( s ¯0 ,n ) = λ a ( a a n − β ( a + a n )) = λ n − β − βλ n . (cid:3) Proof of Theorem 1 . Let V be a primitive 2-generated symmetric axial algebra ofMonster type (2 , ) over a field of characteristic 5 such that λ = λ = 1. By [1,Lemma 4.4], for h ∈ Z , the ad a -eigenvectors u h and v h , defined in Section 1, areas follows (remind that = − u h = − a + ( a h + a − h ) + 2 s ¯0 ,h , (7) v h = a + 2( a h + a − h ) − s ¯0 ,h . By the fusion law, for every h, k ∈ Z , the following identities hold(8) a ( u h u k − v h v k + λ a ( v h v k ) a ) = 0 -GENERATED AXIAL ALGEBRAS 7 and(9) a ( u h u k + u h v k ) = 2 u h v k . By [1, Lemma 4.3], for every j ∈ Z , we have a s ¯0 ,j = − a + ( a − j + a j ) − s ¯0 ,j .Using the action of the group of automorphisms h τ , f i , we get, for every j, k ∈ Z , r ∈ Z ,(10) a r + jk s ¯ r,j = − a r + jk + ( a r + j ( k − + a r + j ( k +1) ) − s ¯0 ,j . Set V := h a i , s ¯ r,n | i ∈ Z , n ∈ Z + , r ∈ Z i and, for t ∈ Z , denote by [ t ] thecongruence class t + 3 Z , 0 ∗ [ t ] := 0, ( − ∗ [1] := −
1, and ( − ∗ [2] := 1. Claim.
For every i ∈ Z + , r ∈ Z , and t ∈ { , , } (i) λ i = 1 and λ a ( s ¯ r,i ) = 0 ;(ii) if i , s ¯ r,i = s ¯0 ,i ;(iii) if i ≡ and r ≡ t , s ¯ r,i = s ¯ t,i ;(iv) for every l ∈ Z , j ≤ i , m ∈ Z , the products a l s ¯ m,j belong to V and satisfythe formula (11) a l s ¯ m,j = − a l +( a l − j + a l + j ) − s ¯ m,j +( δ [ l ] [ m ] − ∗ [ l − m ] ( s ¯ m − ¯1 ,j − s ¯ m +¯1 ,j ) . Note that, by the symmetries of V , part (iv) of Claim holds if and only if, forevery r ∈ { , . . . , j − } , the products a s ¯ r,j satisfy the corresponding formula inEquation (11).We proceed by induction on i . Let i = 1. By the hypothesis λ = 1, hence (i)holds by Lemma 7.(4) and (ii) holds trivially.Let i = 2. Again by the hypothesis, λ = 1, hence (i) holds by Lemma 7.(4).Equation (1) in [1, Lemma 4.8] becomes − s ¯0 , − s ¯1 , ) = 0, whence(12) s ¯1 , = s ¯0 , and parts (ii) and (iv) hold.Assume i ≥ l ≤ i . By the fusion law, u u i , u v i are 0- and 2-eigenvectors for ad a , respectively. Further, since( s ¯1 ,i +1 ) τ = s − ¯1 ,i +1 = s ¯ ı,i +1 , we get that s ¯1 ,i +1 − s ¯ ı,i +1 is negated by the map τ , in particular s ¯1 ,i +1 − s ¯ ı,i +1 isa − a . It follows that λ a ( u u j ) = λ a ( u v j ) = λ a ( s ¯1 ,i +1 − s ¯ ı,i +1 ) = 0 . By Equations (6) and (7) and linearity of λ a , we get0 = λ a ( u u i + u v i ) = 2 − λ a ( s ¯1 ,i +1 ) − λ i +1 , and 0 = λ a ( u u i ) = λ a ( s ¯0 , s ¯0 ,i ) − λ a ( s ¯1 ,i +1 ) + 2 λ i +1 − , whence(13) λ a ( s ¯1 ,i +1 ) = 1 − λ i +1 and λ a ( s ¯0 , s ¯0 ,i ) = λ i +1 − . As above, substituting u , u i , v , and v i in Equation (8), with h = 1 and k = i , weget(14) a ( s ¯1 ,i +1 + s ¯ ı,i +1 ) = − λ i +1 ) a + 2( a i +1 + a − i − ) − s ¯0 ,i +1 . On the other hand, since s ¯1 ,i +1 − s ¯ ı,i +1 is a − a ,(15) a ( s ¯1 ,i +1 − s ¯ ı,i +1 ) = − s ¯1 ,i +1 − s ¯ ı,i +1 ) . CLARA FRANCHI AND MARIO MAINARDIS
Taking the sum and the difference of both members of Equations (14) and (15) weget a s ¯1 ,i +1 = − (1 + λ i +1 ) a + ( a i +1 + a − i − ) − s ¯0 ,i +1 − s ¯1 ,i +1 + s ¯ ı,i +1 (16)and a s ¯2 ,i +1 = − (1 + λ i +1 ) a + ( a i +1 + a − i − ) − s ¯0 ,i +1 + s ¯1 ,i +1 − s ¯ ı,i +1 . (17)Assume first that i + 1 ≡
0. Substituting the expressions (6) and (7) in Equa-tion (9), with h = 1 and k = i , and using Equation (14) we get(18) s ¯0 , s ¯0 ,i = 2( λ i +1 − a − s ¯1 ,i +1 + s ¯ ı,i +1 + s ¯0 ,i +1 ) + 2 s ¯0 , − s ¯0 ,i − + 2 s ¯0 ,i . Since, s ¯0 , and, by the inductive hypothesis, s ¯0 ,i − and s ¯0 ,i are σ j -invariant forevery j ∈ Z , subtracting to both members of Equation (18) their images under σ i +1 we get(19) 0 = s ¯0 , s ¯0 ,i − ( s ¯0 , s ¯0 ,i ) σ i +1 = 2( λ i +1 − a − a i +1 ) , whence either λ i +1 = 1 or a = a i +1 . But, again, in the latter case, λ i +1 = λ a ( a ) = 1. Hence, by Equation (13), giving (i). In particular Equation (18)becomes(20) s ¯0 , s ¯0 ,i = − s ¯1 ,i +1 + s ¯ ı,i +1 + s ¯0 ,i +1 ) + 2 s ¯0 , − s ¯0 ,i − + 2 s ¯0 ,i . Again, taking the image under σ i of both members of the above equation, we get0 = s ¯0 , s ¯0 ,i − ( s ¯0 , s ¯0 ,i ) σ i = − s ¯1 ,i +1 − s ¯ ı − ¯1 ,i +1 ) , whence s ¯1 ,i +1 = s ¯ ı − ¯1 ,i +1 . Since, by Lemma 7.(3), the group h τ , f i is transitive onthe set { s ¯ r,i +1 | ≤ r ≤ i } , it follows that (iii) holds, in particular(21) s ¯ ı,i +1 = s ¯2 ,i +1 . Hence, in order to prove (iv), we just need to check that Equation (11) holdsfor l = 0 and m ∈ { , , } . The case m = 0 follows from Equation (10), cases m ∈ { , } follow from Equations (16), (17), and (21).Assume now i + 1 ≡
1. Substituting the expressions (6) and (7) in Equation (9),with h = 2 and k = i −
1, since by the inductive hypothesis (iv), for every l ∈ Z and j ≤ i + 1, r ∈ Z , the products a l s ¯ r,j are given by Equation (11), we get s ¯0 , s ¯0 ,i − = 2( λ i +1 − a − s ¯1 ,i − + s ¯2 ,i − + s ¯0 ,i − ) + 2 s ¯0 , − s ¯0 ,i +1 + 2 s ¯0 ,i − . Since, s ¯0 , and, by the inductive hypothesis, s ¯0 ,i − and s ¯1 ,i − + s ¯2 ,i − + s ¯0 ,i − are σ j -invariant for every j ∈ Z , as above we obtain(22) 0 = s ¯0 , s ¯0 ,i − − ( s ¯0 , s ¯0 ,i − ) σ i +1 = 2( λ i +1 − a − a i +1 ) , whence, as in the previous case, λ i +1 = 1 giving (i) by Equation (13). Similarly,0 = s ¯0 , s ¯0 ,i − − ( s ¯0 , s ¯0 ,i − ) σ = s ¯1 ,i +1 − s ¯0 ,i +1 , whence s ¯1 ,i +1 = s ¯0 ,i +1 . Since the group h τ , f i is transitive on the set of all pairs( s ¯ r,i +1 , s ¯ r +¯1 ,i +1 ), it follows that s ¯ r,i +1 = s ¯0 ,i +1 , for every r ∈ Z .Finally, assume i + 1 ≡
2. Substituting the expressions (6) and (7) in Equa-tion (9), with h = 1 and k = i , using the inductive hypothesis as above (in particular s ¯ i − ,i − = s ¯2 ,i − ), we get s ¯0 , s ¯0 ,i = 2( λ i +1 − a − s ¯0 ,i − + s ¯1 ,i − + s ¯2 ,i − ) + 2 s ¯0 , − s ¯0 ,i +1 + 2 s ¯0 ,i . -GENERATED AXIAL ALGEBRAS 9 Since s ¯0 , and, by the inductive hypothesis, s ¯0 ,i and s ¯0 ,i − + s ¯1 ,i − + s ¯2 ,i − are σ j -invariant for every j ∈ Z , we have(23) 0 = s ¯0 , s ¯0 ,i − ( s ¯0 , s ¯0 ,i ) σ i +1 = 2( λ i +1 − a − a i +1 ) , whence, as above, it follows λ i +1 = 1. Hence, by Equation (13), giving (i). Then,0 = s ¯0 , s ¯0 ,i − ( s ¯0 , s ¯0 ,i ) σ = s ¯1 ,i +1 − s ¯0 ,i +1 , whence we conclude as in the previuous case. This finishes the inductive step andthe Claim is proved. As a consequence, we get that the subspace V is closed withrespect to the multiplication by any axes a i .We now consider the products s ¯ r,i s ¯ t,j , for i, j ∈ Z + , r, t ∈ Z . Proceeding asabove, by Equation (9) with h = i and k = j , we obtain(24) s ¯0 ,i s ¯0 ,j = 2( s ¯0 ,i + s ¯0 ,j ) − s ¯0 , | i − j | + s ¯1 , | i − j | + s ¯2 , | i − j | + s ¯0 ,i + j + s ¯1 ,i + j + s ¯2 ,i + j ) , if { i, j } 6⊆ Z , and(25) s ¯0 ,i s ¯0 ,j = 2( s ¯0 ,i + s ¯0 ,j ) − ( s ¯0 , | i − j | + s ¯0 ,i + j ) , if i ≡ j ≡ . If i
0, then s ¯0 ,i = s ¯1 ,i = s ¯2 ,i , thus, applying f and τ to Equation (24), we get s ¯0 ,i s ¯1 ,j = 2( s ¯0 ,i + s ¯1 ,j ) − s ¯0 , | i − j | + s ¯1 , | i − j | + s ¯2 , | i − j | + s ¯0 ,i + j + s ¯1 ,i + j + s ¯2 ,i + j ) , and s ¯0 ,i s ¯2 ,j = 2( s ¯0 ,i + s ¯2 ,j ) − s ¯0 , | i − j | + s ¯1 , | i − j | + s ¯2 , | i − j | + s ¯0 ,i + j + s ¯1 ,i + j + s ¯2 ,i + j ) . If i ≡ j ≡
0, in a similar way, from Equation (25), we get, for any r ∈ Z , s ¯ r,i s ¯ r,j = 2( s ¯ r,i + s ¯0 r,j ) − ( s ¯ r, | i − j | + s ¯ r,i + j ) . The last products needed are s ¯ r, h s ¯ t, k , for h, k ∈ Z + , r, t ∈ Z with r t . For k ∈ Z + , set(26) u k := a + 2( a − k + a k ) − s ¯0 , k + s ¯1 , k + s ¯2 , k ) . Then, by Equation (11), u k is a 0-eigenvector for ad a . Hence, by the fusion law,we have a ( u k u h + u k v h ) = 2 u k v h . Substituting the expressions (26) and (7) in the above equation, we get(27) s ¯0 , h ( s ¯1 , k + s ¯2 , k ) = − s ¯0 , h − s ¯0 , k − s ¯0 , | h − k | + s ¯0 , h + k ) ) , whence, taking the images under f of both sides, we get(28) s ¯1 , h ( s ¯0 , k + s ¯2 , k ) = − s ¯1 , h − s ¯1 , k − s ¯1 , | h − k | + s ¯1 , h + k ) ) . Taking the difference of the Equations (27) and (28), we obtain s ¯2 , k ( s ¯0 , h − s ¯1 , h ) = − ( s ¯0 , h − s ¯1 , h + s ¯0 , k − s ¯1 , k )(29) − s ¯0 , | h − k | − s ¯1 , | h − k | + s ¯0 , h + k ) − s ¯1 , h + k ) ) . Since in Equation (29) we can swap h and k , we finally get s ¯0 , h s ¯1 , k = 12 (cid:2) s ¯0 , h ( s ¯1 , k + s ¯2 , k ) − ( s ¯2 , h ( s ¯0 , k − s ¯1 , k )) τ (cid:3) = 2( s ¯0 , h + s ¯1 , h − s ¯2 , h + s ¯0 , k + s ¯1 , k − s ¯3 k ) − ( s ¯0 , | h − k | + s ¯1 , | h − k | − s ¯2 , | h − k | + s ¯0 , h + k ) + s ¯1 , h + k ) − s ¯3( h + k ) ) . Using the maps τ and f , we derive the formulas for the products s ¯0 , h s ¯2 , k and s ¯1 , h s ¯2 , k . Hence V is a subalgebra of V , and since a , a ∈ V , we get V = V .Therefore, the map φ : ˆ H → V ˆ a i a i ˆ s ¯ r,j s ¯ r,j from the basis B of ˆ H and V , extends to a surjective linear map ¯ φ : ˆ H → V . ¯ φ isactually an algebra homomorphism, since V satisfies the multiplication table of thealgebra ˆ H and the result follows. (cid:3) Proof of Theorem 2 . Let V be a primitive 2-generated axial algebra of Monstertype ( α, β ) over a field F of characteristic 5, such that D ≥
6. If α = 4 β , condition D ≥ α, β ) = (2 , ) and λ = λ = 1.Then, by Theorem 1, V satisfies (3). If α = 2 β , the proof of Claim 5.18 in [10]is still valid in characteristic 5, proving (2). Similarly, if α = 4 β, β , the proof ofClaim 5.19 in [10] is also valid in characteristic 5 and gives (1). (cid:3) References [1] Franchi, C., Mainardis, M., Shpectorov, S., 2-generated axial algebras of Monster type ( α, β ).In preparation.[2] Franchi, C., Mainardis, M., Shpectorov, S., An infinite dimensional 2-generated axial algebraof Monster type. https://arxiv.org/abs/2007.02430. [3] Galt, A., Joshi, V., Mamontov, A., Shpectorov, S., Staroletov, A., Double axes and subalge-bras of Monster type in Matsuo algebras. https://arxiv.org/abs/2004.11180. [4] Hall, J., Rehren, F., Shpectorov, S.: Universal Axial Algebras and a Theorem of Sakuma,
J. Algebra (2015), 394-424.[5] Ivanov, A. A.: The Monster group and Majorana involutions. Cambridge Tracts in Mathe-matics 176, Cambridge Univ. Press, Cambridge (2009)[6] Rehren, F., Axial algebras, PhD thesis, University of Birmingham, 2015.[7] Rehren, F., Generalised dihedral subalgebras from the Monster,
Trans. Amer. Math. Soc. (2017), 6953-6986.[8] Nadirashvili, N., Tkachev, Vladimir G., Vladut, S., Nonlinear Elliptic Equations and Nonas-sociative Algebras Math. Surveys and Monographs, vol. 200, AMS, Providence, RI (2014)[9] Tkachev, Vladimir G., Spectral properties of nonassociative algebras and breaking regularityfor nonlinear elliptic type PDEs,
Algebra I Anal. (2) (2019), 51-74.[10] Yabe, T.: On the classification of 2-generated axial algebras of Majorana type. https://arxiv.org/abs/2008.01871 Dipartimento di Matematica e Fisica, Universit`a Cattolica del Sacro Cuore, ViaMusei 41, I-25121 Brescia, Italy
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