2-generated axial algebras of Monster type
aa r X i v : . [ m a t h . R A ] J a n -GENERATED AXIAL ALGEBRAS OF MONSTER TYPE CLARA FRANCHI, MARIO MAINARDIS, SERGEY SHPECTOROV
Abstract.
In this paper we provide the basic setup for a project, initiatedby Felix Rehren in [16], aiming at classifying all 2-generated primitive axialagebras of Monster type ( α, β ). We first revise Rehren’s construction of aninitial object in the cathegory of primitive n -generated axial algebras giving aformal one, filling some gaps and, though correcting some inaccuracies, confirmRehren’s results. Then we focus on 2-generated algebras which naturally partinto three cases: the two critical cases α = 2 β and α = 4 β , and the generic case(i.e. all the rest). About these cases, which will be dealt in detail in subsequentpapers, we give bounds on the dimensions (the generic case already treated byRehen) and classify all 2-generated primitive axial algebras of Monster type( α, β ) over Q ( α, β ) for α and β algebraically independent indeterminates over Q . Finally we restrict to the 2-generated Majorana algebras (i.e. when α = and β = ), showing that these fall precisely into the nine isomorphism typesof the Norton-Sakuma algebras. Introduction
Axial algebras constitute a class of commutative non-associative algebras gener-ated by idempotent elements called axes such that their adjoint action is semisim-ple and the relative eigenvectors satisfy a prescribed fusion law. Let R be a ring, { α, β } ⊆ R \ { , } and α = β . An axial algebra over R is called of Monster type ( α, β ) if it satisfies the fusion law M ( α, β ) given in Table 1. This means that theadjoint action of every axis has spectrum { , , α, β } and, for any two eigenvectors v γ , v δ with relative eigenvalues γ, δ ∈ { , , α, β } , the product v γ · v δ is a sum ofeigenvectors relative to eigenvalues contained in γ ⋆ δ . This class was introduced byJ. Hall, F. Rehren and S. Shpectorov [7] in order to axiomatise some key features ofimportant classes of algebras, including the weight-2 components of OZ-type vertexoperator algebras, Jordan algebras and Matsuo algebras (see the introductions of[7], [16] and [5]). They are also of particular interest for finite group theorists asmost of the finite simple groups, or their automorphism groups, can be faithfullyand effectively represented as automorphism groups of these algebras. ⋆ α β ∅ α β ∅ α βα α α , ββ β β β , , α Table 1.
Fusion table M ( α, β ) The motivating example is the Griess algebra V ♯ which is a real axial algebraof Monster type ( , ) and coincides with the weight-2 component of the Monstervertex operator algebra V ♯ . Here axes are associated to the involutions of type 2 A inthe Monster (i.e. those having the double cover of the Baby Monster as centraliser).The subalgebras of the Griess algebra which are generated by two axes were firstclassified by S. Norton in [14] who showed that there are nine isomorphism classesof such algebras, corresponding to the 9 conjugacy classes in the Monster group M of the dihedral subgroups generated by the pairs of 2 A involutions associated tothe two generating axes. These algebras are labelled 1 A , 2 A , 3 A , 4 A , 5 A , 6 A , 4 B ,2 B .On the basis of an earlier work by M. Miyamoto [12], who observed that Isingvectors in a vertex operator algebra of CFT-type satisfy the Monster fusion law,in [17] S. Sakuma classified all OZ -type vertex operator algebras generated by apair of two Ising conformal vectors showing that, up to rescaling, the isomorphismtypes of their weight-2 subspaces match precisely the 9 classes of Norton, now oftencalled Norton-Sakuma algebras.By extracting the relevant properties of these weight-2 subspaces, A. A. Ivanovintroduced in 2009 the concept of Majorana algebras [10], which are real axial al-gebras of Monster type ( , ) satisfying some additional properties, in particularthey are endowed with a positive definite associative bilinear form. In 2010 A.Ivanov, D. Pasechnik, A. Seress and S. Shpectorov obtained Norton’s classificationwithin the axiomatic context of Majorana algebras (see [11]). A further develop-ment was achieved by J. Hall, F. Rehren and S. Shpectorov [7] who constructeda universal object for primitive Frobenius axial algebras with a prescribed fusionlaw and extended Norton-Sakuma theorem to 2-generated primitive Frobenius ax-ial algebras of Monster type ( , ). Subsequently F. Rehren [16, 15] addressedthe general case dropping the assumption on the existence of the bilinear form(which characterises Frobenius axial algebras) and described a universal object forprimitive axial algebras with a prescribed fusion law. In the particular case of 2-generated algebras of Monster type ( α, β ) with α
6∈ { β, β } and in characteristicother than 2, he produced a spanning set of 8 elements and computed the struc-ture constants with respect to these elements. Finally he produced new examplesof 2-generated primitive Frobenius axial algebras of Monster type. This paper ispart of a project of the authors aiming to classifying 2-generated primitive axialalgebras of Monster type over fields. We start by giving, for any positive integer n ,a formal construction of a universal n -generated primitive axial algebra mappingepimorphically onto every n -generated axial algebra with a prescribed fusion law.We then focus on 2-generated primitive axial algebras of Monster type. We saythat such an algebra is symmetric if the map that swaps the two generating axesextends to an automorphism of the entire algebra. All the algebras considered byRehren in [15, 16] are symmetric. In Section 4, we re-prove Rehren’s result on thenumber of generators and get the following bound for symmetric algebras in thecase α = 4 β . Theorem 1.1.
Every -generated primitive axial algebra of Monster type ( α, β ) over a field F of characteristic other than has dimension at most , provided α
6∈ { β, β } . Theorem 1.2.
Every -generated symmetric primitive axial algebra of Monstertype (4 β, β ) over a field F of characteristic other than has dimension at most ,except possibly when ( α, β ) = (2 , ) . The case ( α, β ) = (2 , ) is truly exceptional, as the infinite dimensional 2-generated symmetric primitive axial algebra of Moster type (2 , ) constructed in [4]shows. On the other hand, the same bound 8 holds for 2-generated primitive axialalgebras of Monster type (2 β, β ) over any ring in which 2 and β are invertible andit is the best possible (see [3]).In Section 5 we consider in more details the case when α − β and α − β areinvertible in the field F , which we call the generic case . Denote by F the primesubfield of F and let F ( α, β )[ x, y, z, t ] be the polynomial ring in 4 variables over F ( α, β ). We prove the following result. Theorem 1.3.
There exists a subset T ⊆ F ( α, β )[ x, y, z, t ] of size , dependingonly on F , α , and β , such that every -generated primitive axial algebra of Mon-ster type ( α, β ) over a field F of characteristic other than , with α, β ∈ F and α
6∈ { β, β } is completely determined, up to homomorphic images, by a quadruple ( x , y , z , t ) ∈ F which is a common zero of all the elements of T . Using Theorem 1.3, we classify the algebras defined over the field Q ( α, β ) with α and β independent indeterminates over Q . We refer to [8] for the definition ofthe algebras of type 1 A , 2 B , 3 C ( η ), η ∈ F . We denote by 3 A ( α, β ) the algebra ofdimension 4 defined in [16, Table 9] for α = . Theorem 1.4.
Let V be a -generated primitive axial algebra of Monster type ( α, β ) over the field Q ( α, β ) , with α and β algebraically independent indeterminatesover Q . Then we have one of the following:(1) V is the trivial algebra Q ( α, β ) of type A ;(2) V is an algebra of type B ;(3) V is an algebra of Jordan type α of type C ( α ) ;(4) V is an algebra of Jordan type β of type C ( β ) ;(5) V is an algebra of dimension of type A ( α, β ) .In particular, V is symmetric. Recall that the fusion law M ( α, β ) admits a Z -grading M ( α, β ) + = { , , α } and M ( α, β ) − = { β } and this implies that every axis induces an automorphismof the algebra called Miyamoto involution. The Miyamoto group is the groupgenerated by all Miyamoto involutions (see [9]). Corollary 1.5.
Let V be a primitive finitely generated axial algebra of Monstertype ( α, β ) over the field Q ( α, β ) , with α and β independent indeterminates over Q . Then the Miyamoto group of V is a group of -transpositions. Finally, as a consequence of Theorem 1.3, we get that the Norton-Sakuma Theo-rem holds in general for 2-generated primitive axial algebra of Monster type ( , )over a field of characteristic zero, without any assumption on the existence of aFrobenius form. This fact has been also checked computationally in [19]. Theorem 1.6.
Every -generated primitive axial algebra of Monster type ( , ) over a field of characteristic zero is a Norton-Sakuma algebra. CLARA FRANCHI, MARIO MAINARDIS, SERGEY SHPECTOROV
Throughout this paper R is a commutative associative ring with 1.While this paper was in preparation, Yabe posted a preprint in arXiv giving analmost complete classification of the primitive symmetric 2-generated axial algebrasof Monster type ( α, β ) [18].2. Primitive axial algebras
We begin with some basic results about endomorphisms of R -modules, whichare well known for vector spaces. The main difference is that, when considering R -modules instead of vector spaces, it is no longer true in general that eigenvectorsrelative to different eigenvalues are linearly independent.Let V be an R -module. For ξ ∈ End ( V ), λ ∈ R , and Γ ⊆ R , define V ξλ := { v ∈ V | ξ ( v ) = λv } and V ξ Γ := X λ ∈ Γ V ξλ . If V is also an R -algebra and a ∈ V , denote by ad a the endomorphism of V inducedby multiplication by a : ad a : V → Vx ax . In this case, we’ll write simply V aλ and V a Γ instead of V ad a λ and V ad a Γ , respectively.Two elements α and β of R are called distinguishable if α − β is a unit in R . In theremainder of this section we assume that Γ is a finite set of pairwise distinguishableelements of R . Note that every nontrivial ring homomorphism maps sets of pairwisedistinguishable elements into sets of pairwise distinguishable elements. Let R [ x ] bethe ring of polynomials over R with indeterminate x . Lemma 2.1. If g ∈ R [ x ] satisfies g ( λ ) = 0 for every λ ∈ Γ and deg g < | Γ | , then g = 0 .Proof. We proceed by induction on | Γ | . If | Γ | = 1, then g is a constant and sincethe value of g is zero in at least one element (the element of Γ), it must be g = 0.Suppose | Γ | >
1. Assume by contradiction that g is not the zero polynomialand set k := deg g . Then k < | Γ | . Let λ ∈ Γ, then, since x − λ is monic, wehave g = q · ( x − λ ) + r , with r = g ( λ ) = 0, so g = q · ( x − λ ). Then clearlydeg q = k − < | Γ ′ | , where Γ ′ = Γ \ { λ } . Also for µ ∈ Γ ′ , 0 = g ( µ ) = q ( µ ) · ( µ − λ ).Since µ and λ are distinguishable, ( µ − λ ) is a unit and so q ( µ ) = 0. By inductionthis means that q = 0, whence also g = 0, a contradiction. (cid:3) For µ ∈ Γ, define(1) f µ := Y λ ∈ Γ \{ µ } ( x − λ ) , and(2) f := Y λ ∈ Γ ( x − λ ) , clearly f = ( x − µ ) f µ for every µ ∈ Γ. Note that, since elements of Γ are pairwise distinguishable, f µ ( µ )is a unit in R . Corollary 2.2. X µ ∈ Γ f µ ( µ ) f µ = 1 . Proof.
Define g := X µ ∈ Γ f µ ( µ ) f µ − , then deg g < | Γ | and clearly g ( λ ) = 0 for every λ ∈ Γ. Hence, by Lemma 2.1, g = 0. (cid:3) Lemma 2.3.
For every ξ ∈ End ( V ) , the following statements are equivalent:(1) V = L λ ∈ Γ V ξλ ;(2) V = V ξ Γ ;(3) f ( ξ ) V = 0 .Proof. Clearly (1) implies (2) and (2) implies (3) . Suppose (3) is satisfied. Then,for every µ ∈ Γ, and v ∈ V ,0 = ( f ( ξ ))( v ) = Y λ ∈ Γ ( ξ − λ ) ! ( v )= ( ξ − µ ) Y λ ∈ Γ \{ µ } ( ξ − λ ) ( v )= ( ξ − µ )(( f µ ( ξ ))( v )) , whence ( f µ ( ξ ))( v ) ∈ V ξµ . Set v µ := 1 f µ ( µ ) ( f µ ( ξ ))( v ) . By Corollary 2.2, id V = X µ ∈ Γ f µ ( µ ) f µ ( ξ )and so v = X µ ∈ Γ f µ ( µ ) ( f µ ( ξ ))( v ) = X µ ∈ Γ v µ . Now, assume v = P µ ∈ Γ w µ for some w µ ∈ V ξµ . Since1 f µ ( µ ) ( f µ ( ξ ))( w λ ) = δ λµ w µ (where δ λµ is the Kronecker delta), we get v µ = 1 f µ ( µ ) ( f µ ( ξ ))( v ) = X µ ∈ Γ f µ ( µ ) ( f µ ( ξ ))( w λ ) = w µ , giving (1) . (cid:3) Recall [2] that a fusion law is a pair ( S , ∗ ) such that S is a set and ∗ is a mapfrom the cartesian product S × S to the power set 2 S . A morphism between twofusion laws ( S , ∗ ) and ( S , ∗ ) is a map φ : S → S such that, for α, β ∈ S , φ ( α ∗ β ) ⊆ φ ( α ) ∗ φ ( β ) . An isomorphism of fusion laws is a bijective morphism such that its inverse is alsoa morphism. A fusion law ( S , ∗ ) is said to be finite if S is a finite set. In this paperwe deal with fusion laws ( S , ∗ ) where S is a finite set containing the spectrumof the adjoint action of an idempotent element in an R -algebra. Therefore, weassume 1 R ∈ S ⊆ R . Without loss of generality, we may also assume that 0 R ∈ S .Further, for every morphism φ of fusion laws, we’ll assume that 1 φ = 1 and 0 φ = 0.More generally, when considering morphisms between fusion laws, one may wantto preserve some possible algebraic relations between the elements of the set S .To this aim, we call a morphism φ of fusion laws an algebraic morphism if it is a Z -linear map. An axial algebra over R with generating set A and fusion law ( S , ⋆ )is a quadruple ( R, V, A , ( S , ⋆ ))such that(1) R is an associative commutative ring with identity 1;(2) S is a subset of R containing 1 and 0;(3) V is a commutative non associative R -algebra;(4) A is a set of idempotent elements (called axes ) of V that generate V as an R -algebra and such thatAx1 V = V a S for every a ∈ A ;Ax2 V aλ V aµ ⊆ V aλ⋆µ for every λ, µ ∈ S and a ∈ A .Further, V is called primitive if,Ax3 V a = Ra for every a ∈ A .A Frobenius axial algebra is an axial algebra (
R, V, A , ( S , ⋆ )) endowed with anassociative bilinear form κ : V × V → R such that the map a κ ( a, a ) is constanton the set of axes.Let ( R, V, A , ( S , ⋆ )) be an axial algebra and assume the elements of S are pairwisedistinguishable. As in the proof of Lemma 2.3, for every v ∈ V , denote by v theprojection of v into V a with respect to the decomposition of V into ad a -eigenspaces.If V is primitive, we have v = λ a ( v ) a for some λ a ( v ) ∈ R which is generally not unique. On the other hand, if theannihilator ideal Ann R ( a ) := { r ∈ R | ra = 0 } of a in R is trivial, then λ a ( v ) is unique, and we say that a is a free axis. Clearlythis condition is satisfied when R is a field. As an immediate consequence we havethe main result of this section. Proposition 2.4.
Let ( R, V, A , ( S , ⋆ )) be a primitive axial algebra and assume thatthe elements of S are pairwise distinguishable and the axes in A are free. Then,for every a ∈ A , there is a well defined R -linear map (3) λ a : V → Rv λ a ( v ) such that every v ∈ V decomposes uniquely as (4) v = λ a ( v ) a + X µ ∈S\{ } v µ , with v µ ∈ V aµ . We say that V is weak primitive if, for every a ∈ A and every element v ∈ V , v can be decomposed as(5) v = l a ( v ) a + X µ ∈S\{ } v µ where l a ( v ) ∈ R depends on v and a , and v µ ∈ V aµ . Note that in general thedecomposition in (5) and l a ( v ) are not uniquely determined by v . Lemma 2.5.
If the elements of S are pairwise distinguishable, in particular, if R is a field, then weak primitivity is equivalent to primitivity.Proof. Trivially, primitivity implies weak primitivity. Conversely, suppose that V is weak primitive, fix a ∈ A and let v ∈ V a . Then, by weak primitivity, there exist l ∈ R , v µ ∈ V aµ , such that v = la + X µ ∈S\{ } v µ . Hence X µ ∈S\{ } v µ = v − la ∈ ( X µ ∈S\{ } V aµ ) ∩ V a , and the last intersection is trivial by Lemma 2.3. Thus v = la ∈ Ra . (cid:3) We conclude this section with the following straightforward observation.
Lemma 2.6.
Let ( R, V, A , ( S , ⋆ )) be a primitive axial algebra and assume that theelements of S are pairwise distinguishable, and the axes in A are free. Then, forevery a ∈ A , γ, δ ∈ S , and v, w ∈ V , we have ( f (ad a ))( w − λ a ( w ) a ) = 0 and Y η ∈ γ⋆δ (ad c − η ) (( f γ ( ad c ))( v ) · ( f δ ( ad c ))( w )) = 0 . Universal primitive axial algebras
In this section we fix a positive integer k , a fusion law ( S , ∗ ), and denote by O the class whose objects are the primitive axial algebras( R, V, A , ( S , ∗ ))such thatO1 A has size at most k and its elements are free axes,O2 ( S , ∗ ) is isomorphic to ( S , ∗ ), andO3 the elements of S are pairwise distinguishable in R . CLARA FRANCHI, MARIO MAINARDIS, SERGEY SHPECTOROV
For two elements(6) V := ( R , V , A , ( S , ∗ )) and V := ( R , V , A , ( S , ∗ ))in O , let H om ( V , V ) be the set of maps φ : R ∪ V → R ∪ V satisfying the following conditions:H1 φ | R is an homomorphism of rings with identity between R and R thatinduces by restriction an isomorphism of fusion laws between ( S , ∗ ) and( S , ∗ );H2 φ | V is a (non-associative) ring homomorphism between V and V such that φ ( A ) ⊆ A ;H3 ( γv ) φ = γ φ v φ , for every γ ∈ R and v ∈ V .Note that, since φ | R is a ring homomorphism, the induced isomorphism of fusionlaw in (H1) is in fact an algebraic isomorphism. Denote by H the class of all φ ∈ H om ( V , V ) where V and V range in O . Then clearly ( O , H ) is a category.It will turn convenient to define some subcategories of ( O , H ) in the followingway. Let- n := |S \ { , }| ;- x i , y i , w i , z i,j , t h be algebraically independent indeterminates over Z , for i, j ∈ { , . . . , n } , with i < j , h ∈ N ;- D be the polynomial ring Z [ x i , y i , w i , z i,j , t h | i, j ∈ { , . . . n } , i < j, h ∈ N ];- L be a proper ideal of D containing the setΣ := { x i y i − , (1 − x i ) w i − , and ( x i − x j ) z i,j − ≤ i < j ≤ n } ;- ˆ D := D/L . For d ∈ D , we denote the element L + d by ˆ d .The extra indeterminates t h in the definition of D have been introduced here inorder to guarantee, when necessary, the invertibility of certain elements in the ringˆ D .Since L is proper and, for 1 ≤ i < j ≤ n , the elements ˆ x i − ˆ x j are invertible inˆ D , the elements ˆ x , . . . , ˆ x n are still pairwise distinguishable in ˆ D . Define ( O L , H L )as the full subcategory of ( O , H ) whose objects are the primitive axial algebras( R, V, A , ( S , ∗ )) ∈ O that satisfy the further conditionO4 R is a ring containing a subring isomorphic to a factor of ˆ D .Clearly, if (Σ) is the ideal of D generated by the set Σ, then O (Σ) = O . Lemma 3.1.
Let V , V ∈ O L be as in Equation (6) and let φ ∈ H om ( V , V ) .Then, for every a ∈ A , v ∈ V , we have λ a φ ( v φ ) = ( λ a ( v )) φ . Proof.
Since V , V are primitive axial algebras, the result follows applying φ to thedecomposition of v in Equation (4). (cid:3) We construct a universal object for each category ( O L , H L ) as follows. Let- A be a set of size k ;- W be the free commutative magma generated by the elements of A subjectto the condition that every element of A is idempotent; - ˆ R := ˆ D [Λ] be the ring of polynomials with coefficients in ˆ D and indetermi-nates set Λ := { λ c,w | c ∈ A , w ∈ W, c = w } where λ c,w = λ c ′ ,w ′ if and onlyif c = c ′ and w = w ′ .- ˆ V := ˆ R [ W ] be the set of all formal linear combinations P w ∈ W γ w w of theelements of W with coefficients in ˆ R , with only finitely many coefficientsdifferent from zero. Endow ˆ V with the usual structure of a commutativenon associative ˆ R -algebra;- ˆ S be the set { , , ˆ x , . . . , ˆ x n } .Let ⋆ : ˆ S × ˆ S → ˆ S be a map such that ( ˆ S , ⋆ ) is a fusion law isomorphic to ( S , ∗ ).Since, obviously, a fusion law is isomorphic to ( S, ∗ ) if and only if it is isomorphic to( ˆ S , ⋆ ), we may assume ( S , ∗ ) = ( ˆ S , ⋆ ). For µ ∈ ˆ S , let f µ be the polynomial definedin Equation (1), for every c ∈ A , let λ c,c := 1, and let J be the ideal of ˆ V generatedby all the elements(7) ( f (ad c ))( w − λ c,w c ) for all c ∈ A and w ∈ W and(8) Y η ∈ γ⋆δ (ad c − η id ˆ V ) (( f γ ( ad c ))( v ) · ( f δ ( ad c ))( w ))for all v, w ∈ ˆ V , γ, δ ∈ ˆ S , c ∈ A . Let I be the ideal of ˆ R generated by all theelements(9) X w ∈ W γ w λ c,w for all c ∈ A , w ∈ W, γ w ∈ ˆ R such that X w ∈ W γ w w ∈ J. Finally, set- A /J := { c + J | c ∈ A} - J := J + I ˆ V ,- R := ˆ R/I ,- V := ˆ V /J ,- α i := x i + I , for i ∈ { , . . . , n } - ¯ c := c + J , for c ∈ A ,- A := { ¯ c | c ∈ A} ,- S := { I , I , α i | i ∈ { , . . . , n }} . Remark 3.2.
Since ˆ D ∩ I = { } and S ≤ ˆ DI /I , ( S , ⋆ ) is a fusion law isomorphicto ( S , ∗ ) and the elements of S are pairwise distinguishable. Lemma 3.3. ( ˆ R, ˆ V /J, A /J, ( ˆ S , ⋆ )) and ( ˆ R, V , A , ( ˆ S , ⋆ )) are primitive axial algebrassuch that, for J ∗ ∈ { J, J } , λ c + J ∗ ( w + J ∗ ) = λ c,w for every c ∈ A and w ∈ W .Proof. Clearly ˆ
V /J is generated by A /J and V is generated by A . Let J ∗ ∈ { J, J } and let f be as in Equation (2). By the definition of J , for every a ∈ A , f (ad a ) ˆ V ⊆ J , whence, by Lemma 2.3, ˆ V /J ∗ satisfies condition Ax1. By Equation (8), ˆ V /J ∗ also satisfies the fusion law ( ˆ S , ⋆ ). Furthermore, for every c ∈ A and w ∈ W , since f (ad c )( w − λ c,w c ) ∈ J ∗ , we get w + J ∗ = ( λ c,w c + X µ ∈ ˆ S\{ } w µ ) + J ∗ , where w µ + J ∗ is a µ -eigenvector for ad c + J ∗ , so ˆ V /J ∗ is also weak primitive. ByRemark 3.2 and Lemma 2.5, ˆ V /J ∗ is also primitive. (cid:3) Lemma 3.4.
Let V := ( R, V, A , ( S , ∗ )) be an element of O L , let φ : ˆ R ∪ ˆ V → R ∪ V be a map that satisfies conditions (H1) (with respect to ( ˆ S , ⋆ ) and ( S , ∗ ) ), (H2),and (H3). Then I ⊆ ker φ | ˆ R , J ⊆ ker φ | ˆ V .Proof. By Lemma 2.6, J ⊆ ker φ | ˆ V , so φ induces an ˆ R -algebra homomorphism φ ˆ V /J : ˆ
V /J → V,v + J v φ . Since, by Lemma 3.3, ˆ
V /J is a primitive axial algebra over the ring ˆ R , for every c ∈ A and w ∈ W , we can write w + J = ( λ c,w c + X µ ∈ ˆ S\{ } w µ ) + J, where, for every µ ∈ ˆ S \ { } , w µ + J is a µ -eigenvector for ad c + J . Condition(H3) implies that, for every µ ∈ S , a ∈ A , φ ˆ V /J maps µ -eigenvectors for ad a + J to µ φ -eigenvectors for ad a φ . Thus, if v = P w ∈ W γ w w ∈ J , then v φ = 0, whence0 = λ a φ ( v φ )= λ a φ X w ∈ W γ φw w φ ! = X w ∈ W γ φw λ a φ ( w φ )= X w ∈ W γ φw λ a φ (( λ a,w a + X µ ∈ ˆ S\{ } w µ ) φ )= X w ∈ W γ φw λ a φ (( λ a,w ) φ a φ ) + X w ∈ W γ φw λ a φ ( X µ ∈ ˆ S\{ } w φµ ))= X w ∈ W γ φw ( λ a,w ) φ = ( X w ∈ W γ w λ a,w ) φ . Thus I ⊆ ker φ | ˆ R . Finally, by condition (H3), ( I ˆ V ) φ = I φ V φ = 0 V = 0, whence J ⊆ ker φ | ˆ V . (cid:3) Lemma 3.5.
We have J = ˆ V , in particular |A| = k .Proof. Let R k be the direct sum of k copies of R . Set B := { e , . . . , e k } , where( e , . . . , e k ) is the canonical basis of R k . Then, for every i ∈ { , . . . , k } , e i is anidempotent and R k = Re i ⊕ ker ad e i . Therefore, for every fusion law ( S R , ∗ R ), the 4-tuple( R, R k , B , ( S R , ∗ R ))is obviously a primitive (associative) axial algebra. By the construction of ˆ V , anybijection from A to B extends uniquely to a map φ ˆ V : ˆ V → R k . Let φ : ˆ R ∪ ˆ V → R ∪ R k be the map whose restrictions to ˆ R and ˆ V are the canonical projection on R and φ ˆ V ,respectively. Then φ satisfies the conditions (H1), (H2), and (H3). Therefore, byLemma 3.4, J ⊆ ker φ | ˆ V = ˆ V . Since k = |A| ≥ |A| ≥ |B| = k , we get |A| = k . (cid:3) Theorem 3.6. V := ( R, V , A , ( S , ⋆ )) is a universal object in the category ( O L , H L ) .Proof. Clearly V is an algebra over R generated by the set of idempotents A . Sinceby Lemma 3.3, ( ˆ R, V , A , ( ˆ S , ⋆ )) is a primitive axial algebra, and I ⊆ Ann ˆ R ( V ),we get that ( R, V , A , ( S , ⋆ )) is a primitive axial algebra. Finally, the axes in A arefree, for, if c ∈ A and r ∈ ˆ R are such that rc ∈ J , then there exist j ∈ J, i ∈ I and P w ∈ W γ w w ∈ ˆ V such that rc − i X w ∈ W γ w w ! = j, whence, by the definition of I , r ∈ i X w ∈ W γ w λ c,w ! + I = I . By Lemma 3.5, the elements of ˆ S are pairwise distinguishable in R , whence V ∈ O L .Now assume V := ( R , V , A , ( S , ∗ )) is an object in O L and let ¯ t : A → A a map of sets. Since every non-empty subset of A generates a primitive axialsubalgebra of V with fusion law ( S , ∗ ) and free axes, without loss of generalitywe may assume that ¯ t is surjective. Let t be the composition of ¯ t with the (bijective)projection of A to A . Since W is the free commutative magma generated by theset of idempotents A there is a unique magma homomorphism χ : W → V , inducing the map t : A → A . Since the elements of Λ are alegbraically independentover ˆ D , there is a unique homomorphism of ˆ D -algebrasˆ ψ : ˆ R → R such that, for c ∈ A and w ∈ W \ { c } ,(10) λ ˆ ψc,w = λ c t ( w χ ) , where λ c t is the function defined in Proposition 2.4. Defineˆ χ : ˆ V → V P w ∈ W γ w w P w ∈ W γ ˆ ψw w χ . Then ˆ χ is a ring homomorphism extending t and such that ( γv ) ˆ χ = γ ˆ ψ v ˆ χ for every γ ∈ ˆ R and v ∈ ˆ V . Since, for every c ∈ A , v ∈ ˆ V , and γ ∈ S ,[(ad c − γ id ˆ V )( v )] ˆ χ = (ad c t − γ ˆ ψ id V )( v ˆ χ ) , by Lemma 2.6, it follows that J is contained in ker ˆ χ and so ˆ χ induces a ringhomomorphism φ V : V → V extending t and such that ( γw ) φ V = γ ˆ ψ w φ V for every γ ∈ ˆ R and w ∈ V . As in theproof of Lemma 3.4 we get that I ⊆ ker ˆ ψ . Let φ R : R → R be the homomorphismof ˆ D -algebras induced by ˆ ψ . Then ( φ R , φ V ) ∈ H om ( V , V ) .Since ˆ R = ˆ D [Λ], φ R is completely determined by its values on the elements λ c,w + I . Further, by Equation (10), for every c ∈ A , w ∈ W \ { c } ,( λ c,w + I ) φ R = λ c t (( w + J ) φ V ) , whence φ R is completely determined by the images ( w + J ) φ V , with w ∈ W . Since φ V is a ring homomorphism extending t , such images are uniquely determined,whence the uniqueness of ( φ R , φ V ). (cid:3) Corollary 3.7.
Every permutation σ of the set A extends to a unique automor-phism f σ ∈ H om ( V , V ) . Note that, for a generic object V , the above assertion is false (see e.g. the algebra Q ( η ) constructed in [5, Section 5.3]). We say that V = ( R, V, A , ( S , ∗ )) ∈ O L is symmetric if every permutation σ of the set A extends to a unique automorphism f σ ∈ H om ( V , V ). Corollary 3.8.
Let V := ( R, V, A , ( S , ∗ )) ∈ O L , then(1) V ⊗ R := ( R ⊗ ˆ D R, V ⊗ ˆ D R, A ⊗ ˆ D , ( S ⊗ ˆ D , ⋆ )) ∈ O L ,(2) R is isomorphic to a factor of R ⊗ ˆ D R ,(3) V is isomorphic to a factor of V ⊗ ˆ D R . Remark 3.9.
Note that, with the notation of Corollaries 3.7 and 3.8, f σ ⊗ id R isan automorphism of V ⊗ R . Questions (1) Can we define a variety of axial algebras corresponding to the fusion law ( S , ∗ ) ?(2) Is it true that any ideal I of ˆ R containing I defines an axial algebra?
4. 2 -generated primitive axial algebras of Monster type ( α, β )In this section we keep the notation of Section 3, with k = n = 2, ( S , ∗ ) equalto the Monster fusion law M ( α, β ), and L an ideal of D containing 2 t −
1, so that(the class of) 2 is invertible in ˆ D . In order to simplify notation we’ll also identify α with α and α with β .Let V = ( V, R, A , ( S , ∗ )) ∈ O L and a ∈ A . Let S + := { , , α } and S − := { β } .The partition {S + , S − } of S induces a Z -grading on S which, on turn, induces, a Z -grading { V a + , V a − } on V where V a + := V a + V a + V aα and V a − = V aβ . It follows that, if τ a is the map from R ∪ V to R ∪ V such that τ a | V inverts V aβ and leavesinvariant the elements of V a + and τ a | R is the identity, then τ a is an involutoryautomorphism of V (see [7, Proposition 3.4]). The map τ a is called the Miyamotoinvolution associated to the axis a . By definition of τ a , the element av − βv of V is τ a -invariant and, since a lies in V a + ≤ C V ( τ a ), also av − β ( a + v ) is τ a -invariant. Inparticular, by symmetry, Lemma 4.1.
Let a and b be axes of V . Then ab − β ( a + b ) is fixed by the 2-generatedgroup h τ a , τ b i . Let A := { a , a } and, for i ∈ { , } , let τ i be the Miyamoto involutions asso-ciated to a i . Set ρ := τ τ , and for i ∈ Z , a i := a ρ i and a i +1 := a ρ i . Since ρ is an automorphism of V , for every j ∈ Z , a j is an axis. Denote by τ j := τ a j thecorresponding Miyamoto involution. Lemma 4.2.
For every n ∈ N , and i, j ∈ Z such that i ≡ j mod n we have a i a i + n − β ( a i + a i + n ) = a j a j + n − β ( a j + a j + n ) , Proof.
This follows immediately from Lemma 4.1. (cid:3)
For n ∈ N and r ∈ { , . . . , n − } set(11) s r,n := a r a r + n − β ( a r + a r + n ) . For every a ∈ A , let λ a be as in Proposition 2.4. Lemma 4.3.
For i ∈ { , , } we have a s ,i = ( α − β ) s ,i + [(1 − α ) λ a ( a i ) + β ( α − β − a + 12 β ( α − β )( a i + a − i ) . Proof.
This is [16, Lemma 3.1]. (cid:3)
For i ∈ { , , } , let(12) 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. Lemma 4.4.
With the above notation,(1) u i = α (( λ a ( a i ) − β − αλ a ( a i )) a + ( α − β )( a i + a − i ) − s ,i ) ;(2) v i = α (( β − λ a ( a i )) a + β ( a i + a − i ) + s ,i ) ;(3) w i = ( a i − a − i ) .Proof. (3) follows from the definitions of τ and a i , (2) is just a rearranging of a a i = λ a ( a i ) a + αv i + βw i using Equation (11), and (1) follows rearrangingEquation (12). (cid:3) For i, j ∈ { , , } , set P ij := u i u j + u i v j and Q ij := u i v j − α s ,i s ,j . Lemma 4.5.
For i, j ∈ { , , } we have (13) s ,i · s ,j = α ( a P ij − αQ ij ) . Proof.
Since u i and v j are a 0-eigenvector and an α -eigenvector for ad a , respec-tively, by the fusion rule, we have a P ij = α ( u i · v j ) and the result follows. (cid:3) From now on we assume V = V . Let f be the automorphism of V induced bythe permutation that swaps a with a as defined in Corollary 3.7. For i ∈ N define(14) λ i := λ a ( a i ) . Note that, by Lemma 3.1, for every i ∈ N , we have λ a ( a − i ) = λ i , λ a ( a ) = λ f , and λ a ( a − ) = λ f . Set T := h τ , τ i and T := h τ , f i . Lemma 4.6.
The groups T and T are dihedral groups, T is a normal subgroupof T such that | T : T | ≤ . For every n ∈ N , the set { s ,n , . . . , s n − ,n } is invariantunder the action of T . In particular, if K n is the kernel of this action, we have(1) K = T ;(2) K = T , in particular s f , = s , ;(3) T /K induces the full permutation group on the set { s , , s , , s , } withpoint stabilisers generated by τ K , τ K and f K , respectively. In partic-ular s f , = s , and s τ , = s , .Proof. This follows immediately from the definitions. (cid:3)
Lemma 4.7.
In the algebra V , the following equalities hold: ( α − β ) a s , = β ( α − β )( a − + a )+ (cid:20) − αβλ + 2 β (1 − α ) λ f + β α − αβ − α + 4 β − β ) (cid:21) ( a + a − )+ 1( α − β ) h (6 α − αβ − α + 4 β ) λ + (2 α − α ) λ λ f +2( − α − αβ + α )( α − β ) λ − β ( α − α − β ) λ f − αβ ( α − β ) λ +(4 α β − αβ + 2 β )( α − β ) (cid:3) a + h − αλ − α − λ f + (4 α − αβ − α + 4 β − β ) i s , +2 β ( α − β ) s , and α − β ) s , · s , = β ( α − β ) ( α − β )( a − + a )+ h αβ ( α − β ) λ + 2( − α + 5 α β + α − αβ − αβ + 4 β ) λ f β ( − α β − α + 14 αβ + 7 αβ − β − β ) (cid:3) ( a − + a )+2 h − α + 4 αβ + α − β ) λ + 2 α (1 − α ) λ λ f +2( α + 4 α β − αβ − αβ + 4 β ) λ + 2 αβ ( α − λ f + αβ ( α − β ) λ + β ( − α − α β + 13 αβ + 4 αβ − β − β ) (cid:3) a +4 h α ( α − β ) λ + α ( α − λ f + ( − α β + 10 αβ + αβ − β ) i s , − αβ ( α − β ) s , + 2 β ( α − β )( α − β ) s , . Proof.
By the fusion law,(15) a ( u · u − v · v + λ a ( v · v ) a ) = 0 . Substituting in the left side of (15) the values for u and v given in Lemma 4.4 weget the first equality. The expression for ( α − β ) a ) s , allows us to write explicitlyas a linear combination of a − , a − , a , a , a , s , , s , , s , the vector( α − β )( a P − αQ ) . Thus, the second equality then follows from Equation (13) in Lemma 4.5. (cid:3)
Lemma 4.8.
In the algebra V , the following equalities hold:(1) β ( α − β ) ( α − β )( a − a − ) = c ,(2) β ( α − β ) ( α − β )( a − a − ) = − c τ , where c = ( α − β ) h αβλ − α − α − β ) λ f + β ( − α − αβ − α + 6 β ) i a − + h − α + 4 αβ + α − β ) λ − α ( α − λ λ f +(6 α + 6 α β − α − αβ − αβ + 8 β ) λ + 4 αβ ( α − λ f + 2 αβ ( α − β ) λ + β ( α − β )( − α − αβ + α + 4 β + 2 β ) (cid:3) a + h α ( α − λ λ f + 4(13 α − αβ − α + 2 β ) λ f − αβ ( β − λ +( − α − α β + 2 α + 16 αβ + 2 αβ − β ) λ f − αβ ( α − β ) λ f + β ( α − β )(2 α + 8 αβ − α − β − β ) (cid:3) a +( α − β ) h α − α − β ) λ − αβλ f + β ( α + 5 αβ + α − β ) i a +( α − β ) h α ( α − β + 1)( λ − λ f ) i s , − β ( α − β ) ( s , − s , ) . Proof.
Since s , s , is invariant under f , we have 4( α − β )[ s , s , − ( s , s , ) f ] =0. Then, equality (1) follows from the expression for 4( α − β ) s , s , given inLemma 4.7. By applying τ to equation in (1) we get (2). (cid:3) From the formulae in Lemmas 4.7 and 4.8, it is clear that we have differentpictures according whether α − β and α − β are invertible in R or not. Since weare most concerned with algebras over a field, later we will also assume that α − β and α − β are either invertible or zero. Thus we deal separately with the followingthree cases:(1) The generic case: the ideal L contains the elements t h for h ∈ N , and h ≥ t −
1, ( x − x ) t −
1, and ( x − x ) t −
1. In this case we set O g := O L .(2) The α = 2 β case: the ideal L contains the elements t h for h ∈ N , and h ≥ t −
1, and x − x . In this case we set O β := O L .(3) The α = 4 β case: the ideal L contains the elements t h for h ∈ N , and h ≥ t −
1, and x − x . In this case we set O β := O L .In [3] the case α = 2 β is considered in details: in particular it is shown thatevery 2-generated primitive axial algebra in O β is at most 8 dimensional and this bound is attained. The following result, which can be compared to Theorem 3.7in [16], is a consequence of the resurrection principle [11, Lemma 1.7]. Proposition 4.9.
Let V = ( R, V , A , ( S , ⋆ )) be the universal object in the cathegory O g . Then V is linearly spanned by the set { a − , a − , a , a , a , s , , s , , s , } .Proof. Let U be the linear span in V of the set B := { a − , a − , a , a , a , s , , s , , s , } with coefficients in R . From Lemmas 4.7 and 4.8, since α − β and α − β areinvertible in R , we get that a · s , , a ∈ U. The set B is clearly invariant under the action of τ and since a f − = a , U isalso invariant under f . By Equation (11), a a and a a are contained in U ;by applying alternatively τ and f we get that U contains all the products a i a j for i, j ∈ Z . Similarly, since by Lemma 4.3, for i ∈ { , } , a s ,i ∈ U , and byLemma 4.7, a s , ∈ U , we get that U contains all the products a j s ,i and a j s , for j ∈ Z , i ∈ { , } .It follows that, for i, j ∈ { , } , the expression on the righthand side of Equa-tion (13) is contained in U , whence s ,i · s ,j is contained in U . As U is invariantunder f , we have also s , · s , ∈ U . Finally, with a similar argument, using theidentity a · ( u i · u − + u i · v − ) = α ( u i · v − ) , we can express s ,i · s , as linear combination of elements of B . Hence U is asubalgebra of V , and since V is generated by a and a , we get that U = V . (cid:3) Remark 4.10.
Note that the above proof gives an explicit way to compute thestructure constants of the algebra V relative to the generating set B . This has beendone with the use of GAP [6] . The explicit expressions however are far too long tobe written explicitly here. Corollary 4.11.
Let V = ( R, V , A , ( S , ⋆ )) be the universal object in the cathegory O g . Then, R is generated as a ˆ D -algebra by λ , λ , λ f , and λ f .Proof. Since, for every v ∈ V , λ a ( v ) = ( λ a ( v f )) f , λ a is a linear function,and R = R f , by Proposition 4.9, we just need to show that, for every v ∈{ a − , a − , a , a , a , s , , s , , s , } , λ a ( v ) can be written as a linear combination,with coefficients in ˆ D , of products of λ , λ , λ f , and λ f . By definition we have λ a ( a ) = 1 , λ a ( a ) = λ , and λ a ( a ) = λ . Since τ is an R -automorphism of V fixing a , we get λ a ( a − ) = λ a (( a ) τ ) = λ ,λ a ( a − ) = λ a (( a ) τ ) = λ , and λ a ( s , ) = λ a ( a a − βa − βa ) = λ − β − βλ ,λ a ( s , ) = λ a ( aa − βa − βa ) = λ − β − βλ . Finally, by the fusion law, u u + u v is a sum of a 0 and an α -eigenvector forad a , whence λ a ( u u + u v ) = 0. By Lemma 4.3, we can compute u u + u v and find u u + u v = w + ( α − β )2 α s , , with w ∈ h a − , a − , a , a , a , s , i . So, we can express λ a ( s , ) and we obtain λ a ( s , ) = 2( α − α − β λ − α − α − β λ λ f + (1 − β ) λ + βλ − β. (cid:3) We conclude this section with a similar result for symmetric algebras over a fieldin the case α = 4 β . Note that, in this case, since we are assuming α = β , we alsoassume that the field is of characteristic other than 3. Proposition 4.12.
Let V be a primitive symmetric axial algebra of Monster type (4 β, β ) over a field F of characteristic greater than , generated by two axes a and a . Then V has dimension at most , unless β − and λ a ( a ) = λ a ( a ) = λ a ( a τ a ) = λ a ( a τ a ) = 1 . Proof.
Let V = ( R, V , A , ( S , ⋆ )) be the universal object in the cathegory O β .Since α − β = 2 β is invertible in R , Lemma 4.7 yields that, a · s , is containedin h a − , a − , a , a , a , s , , s , , s , i . Since α = 4 β , Equation (1) in Lemma 4.8becomes0 = (cid:2) (48 β ) λ − (108 β − β ) (cid:3) a − + h ( − β + 8 β ) λ + ( − β + 16 β ) λ λ f + (416 β − β ) λ +(16 β − β ) λ f + (24 β ) λ + ( − β + 18 β ) i a + h (64 β − β ) λ λ f + (128 β − β ) λ f + ( − β + 16 β ) λ (16) +( − β + 32 β ) λ f + ( − β ) λ f + (180 β − β ) i a + h ( − β ) λ f + (108 β − β ) i a +48 β (2 β + 1)( λ − λ f ) s , − β ( s , − s , ) . By Corollary 3.8, V is a homomorphic image of V ⊗ ˆ D F , via a homomorphism φ V mapping a i to a i , for i ∈ { , } and F is a homomorphic image of R ⊗ ˆ D F via φ R . Weuse the bar notation to denote the images of the elements of V ⊗ ˆ D F via φ V , while weidentify the image under φ R of an element of R ⊗ ˆ D F with the element itself. Whenwe apply φ V to the relation (16) we get a similar relation in V . If the coefficient of a is not zero in F , then we get a ∈ U := h a − , a , a , s , , s , , s , i . Since V issymmetric, f induces an automorphism ¯ f of V and U is ¯ f invariant. Since U isalso τ ¯ a -invariant, we get also a − ∈ U . More generally, by applying alternatively τ ¯ a and ¯ f we get that ¯ a i ∈ U for every i ∈ Z . The argument used in the proof ofProposition 4.9 yields V = U . If the coefficient of a in the Equation (16) is zeroin F , then we may consider the coefficient of a − and if it is not zero we deduce asabove that a − ∈ h a , a , s , , s , , s , i . By proceeding as in the previous case, we get V = h a , a , s , , s , , s , i . If the coefficients of a and a − are both zero, thenwe get λ ¯ a (¯ a ) = λ ¯ a (¯ a ) = 18 β − . As above, if the coefficient of a (or the coefficient of a ) is not zero, we can express a (or a respectively) as a linear combination of a − , s , , s , − s , ( a , s , , s , − s , respectively). In both cases, it follows that V = h a − , a , a , s , , s , , s , i . Ifalso the coefficients of a and a are both zero, then we get λ ¯ a (¯ a ) = λ ¯ a (¯ a − ) = 480 β − β + 28 β − β and Equation (16) becomes 0 = 36 β ( s , − s , ) . Hence, since F has caracteristic greater than 3, s , = s , and the identity λ a ( s , ) = λ a ( s , ) gives that β satisfies the relation(17) (2 β − (12 β − β −
1) = 0 . From now on assume β ∈ { , } \ { } , in particular F has characteristic otherthan 5. Set U := h a − , a − , a − , a , a , a , a , s , , s , , s , i . From the identity a ( u u − v v + λ a ( v v ) a ) = 0 we can express a ( s , + s , ) as a linear combi-nation of a − , a − , a − , a , a , a , a , s , , and s , and then, by Lemma 4.5, we getthat s , s , ∈ U . Then, from the identity s , s , − ( s , s , ) f = 0, we derive that s , ∈ U , whence also s , = ( s , ) τ a ∈ U . From the identity s , − ( s , ) f = 0 wethen get a ∈ U . It follows that U is invariant under f and τ a , hence a ± i ∈ U for i ≥
4. Since U is also ad a -invariant, it follows that U contains s r,n for every n ≥ r ∈ { , . . . , n − } . Thus U is a subalgebra of V , whence V = U . From the identity a ( v v − λ a ( v v ) a ) = 0 we get s , ∈ h a − , a − , a − , a , a , a , a , s , , s , i . Fi-nally, from the identity a − a = 0 we get a − ∈ h a − , a − , a , a , a , a , s , , s , i ,that is V has dimension at most 8. (cid:3) The generic case
Let V = ( R, V , A , ( S , ⋆ )) be the universal object in the cathegory O g . Note thatin this case we haveˆ D = Z [1 / , x , x , x − , x − , ( x − x ) − , ( x − x ) − , ( x − x ) − ] . By Corollary 4.11, R = ˆ D [ λ , λ f , λ , λ f ]. The elements λ , λ f , λ , λ f are not nec-essarily indeterminates on ˆ D , as they have to satisfy various relations imposed bythe definition of R . In particular, since by Lemma 4.6, s , − s f , = 0, in the ring R the following relations hold(1) λ a ( s , − ( s , ) f ) = 0,(2) λ a (( s , − ( s , ) f ) τ ) = 0,(3) λ a ( a a − a ) = 0,(4) λ a ( s , − ( s , ) f ) = 0.By Remark 4.10, the four expressions on the left hand side of the above identitiescan be computed explicitly and produce respectively four polynomials p i ( x, y, z, t )for i ∈ { , . . . , } in ˆ D [ x, y, z, t ] (with x, y, z, t indeterminates on ˆ D ), that simul-taneously annihilate on the quadruple ( λ , λ f , λ , λ f ). Define also, for i ∈ { , } , q i ( x, z ) := p i ( x, x, z, z ). The polynomials p i ’s and q i ’s are too long to be displayedhere but can be computed using [1] or [6].Suppose V is a primitive axial algebra of Monster type ( α, β ) over a field F ofodd characteristic , with α, β ∈ F and α
6∈ { β, β } , generated by two axes ¯ a and ¯ a . Then, by Corollary 3.8, V is a homomorphic image of V ⊗ ˆ D F and F is ahomomorphic image of R ⊗ ˆ D F . We denote the images of an element δ of R ⊗ ˆ D F in F by ¯ δ and by p i and q i the polynomials in F [ x, y, z, t ] and F [ x, z ] correspondingto p i and q i , respectively. Set T := { p , p , p , p } and T s := { p ( x, z ) , p ( x, z ) } . Moreover, for P ∈ { T, T s } , denote by Z ( P ) the set of common zeroes of all theelements of P in F and F respectively. It is clear from the definition that the p i ’shave the coefficients in the field F ( α, β ). By Proposition 4.9 and Corollary 4.11, thealgebra V is completely determined, up to homomorphic images, by the quadruple( λ ¯ a (¯ a ) , λ ¯ a (¯ a ) , λ ¯ a (¯ a ) , λ ¯ a (¯ a − )) . Furthermore, this quadruple is the homomorphic image in F of the quadruple( λ , λ f , λ , λ f ) and so it is a common zero of the elements of T .If, in addition, the algebra V is symmetric, then λ ¯ a (¯ a ) = λ ¯ a (¯ a ) and λ ¯ a (¯ a ) , = λ ¯ a (¯ a − ))and the pair ( λ ¯ a (¯ a ) , λ ¯ a (¯ a )) is a common zero of the elements of the set T s .We thus have proved Theorem 1.3.Computing the resultant of the polynomials p ( x, z ) and p ( x, z ) with respect to z one obtains a polynomial in x of degree 10, which is the product of the five linearfactors x, x − , x − α, x − β, α − x − (3 α + 3 αβ − α − β )and a factor of degree at most 5. This last factor has degree 5 and is irreduciblein Q ( α, β ), if α and β are indeterminates over Q . On the other hand, for certainvalues of α and β , this factor can be reducible: for example, it even completelysplits in Q ( α, β )[ x ] when α = 2 β (see [3]), or in the Norton-Sakuma case, when( α, β ) = (1 / , /
32) (see the proof of Theorem 1.6 below).Fixed a filed F , in order to classify primitive generic axial algebras of Monstertype ( α, β ) over F generated by two axes ¯ a and ¯ a we can proceed as follows. Wefirst find all the zeroes of the set T s and classify all symmetric algebras. Then weobserve that, the even subalgebra hh ¯ a , ¯ a ii and the odd subalgebra hh ¯ a − , ¯ a ii aresymmetric, since the automorphisms τ ¯ a and τ ¯ a respectively, swap the generatingaxes. Hence, from the classification of the symmetric case, we know all possiblevalues for the pairs ( λ ¯ a (¯ a ) , λ ¯ a (¯ a − ))and we can look for common zeros ( x , y , z , t ) of the set T with those prescribedvalues for ( x , z ).Using this method, we now classify 2-generated primitive axial algebras of Mon-ster type ( α, β ) over the field Q ( α, β ), with α and β independent indeterminatesover Q . Lemma 5.1. If F = Q ( α, β ) , with α and β independent indeterminates over Q ,the set Z ( T s ) consists exactly of the points (1 , , (0 , , (cid:18) β , β (cid:19) , (cid:16) α , (cid:17) , and ( q ( α, β ) , q ( α, β )) , with q ( α, β ) = (3 α + 3 αβ − α − β )4(2 α − . Proof.
The system can be solved in Q ( α, β ) using [1] giving the five solutions of thestatement. (cid:3) Lemma 5.2.
Let F = Q ( α, β ) , with α and β independent indeterminates over Q and let ( x , z ) ∈ Z ( T s ) . Then ( x , y , z , t ) ∈ Z ( T ) if and only if ( y , t ) = ( x , z ) . Proof.
This have been checked using [1]. (cid:3)
Lemma 5.3.
The algebras C ( α ) , C ( β ) , and A ( α, β ) over the field Q ( α, β ) aresimple.Proof. The claim follows from [9, Theorem 4.11 and Corollary 4.6]. For algebras3 C ( α ) and 3 C ( β ) it is proved in [8, Example 3.4]. The algebra 3 A ( α, β ) is the sameas the algebra 3 A ′ α,β defined by Reheren in [16]. By [16, Lemma 8.2], it admits aFrobenius form wich is non degenerate over the field Q ( α, β ) and such that all thegenerating axes are non-singular with respect to this form. Hence, by Theorem 4.11in [9], every non trivial ideal contains at least one of the generating axes. Then,Corollary 4.6 in [9] yields that the algebra is simple. (cid:3) Proof of Theorem 1.4.
It is straightforward to check that the algebras 1 A , 2 B ,3 C ( α ), 3 C ( β ), and 3 A ( α, β ) are 2-generated symmetric axial algebras of Monstertype ( α, β ) over the field Q ( α, β ) and their corresponding values of ( λ ¯ a (¯ a ) , λ ¯ a (¯ a ))are respectively(1 , , (0 , , (cid:18) β , β (cid:19) , (cid:16) α , (cid:17) , and ( q ( α, β ) , q ( α, β )) , where q ( α, β ) is defined in Lemma 5.1. Let V be an axial algebra of Monster type( α, β ) over the field Q ( α, β ) generated by the two axes ¯ a and ¯ a . Set¯ λ := λ ¯ a (¯ a ) , ¯ λ ′ := λ ¯ a (¯ a ) , ¯ λ := λ a (¯ a ) , and ¯ λ ′ := λ ¯ a (¯ a − ) . By Theorem 1.3, V is determined, up to homomorphic images, by the quadruple(¯ λ , ¯ λ ′ , ¯ λ , ¯ λ ′ ), which must be in Z ( T ). By Lemma 5.2, we get the five quadruples(1 , , , , (0 , , , , (cid:18) β , β , β , β (cid:19) , (cid:16) α , , α , (cid:17) , and ( q ( α, β ) , q ( α, β ) , q ( α, β ) , q ( α, β )) . By Corollary 3.8 and Proposition 4.9, V is linearly generated on Q ( α, β ) by the set¯ a − , ¯ a − , ¯ a , ¯ a , ¯ a , ¯ s , , ¯ s , , and ¯ s , . Define¯ d := ¯ s , − ¯ s τ , , ¯ d := ¯ d f , ¯ d := ¯ d τ , and, for i ∈ { , , } , ¯ D i := ¯ d τ i − ¯ d i . By Lemma 4.6, all vectors ¯ d i , ¯ D i for i ∈ { , , } are zero. For all the admissiblevalues of (¯ λ , ¯ λ ′ , ¯ λ , ¯ λ ′ ), the coefficient of ¯ a − in D is non zero, hence we canexpress ¯ a − as a linear combination of ¯ a − , ¯ a , ¯ a ,¯ a , ¯ s , , ¯ s , , and ¯ s , . Similarly,from identity ¯ d = 0 we can express ¯ s , as a linear combination of ¯ a − , ¯ a , ¯ a ,¯ a ,¯ s , , and ¯ s , .For (¯ λ , ¯ λ ′ , ¯ λ , ¯ λ ′ ) in (cid:26)(cid:18) β , β , β , β (cid:19) , ( q ( α, β ) , q ( α, β ) , q ( α, β ) , q ( α, β )) (cid:27) , from identity ¯ d = 0 we get ¯ a − = ¯ a and consequently ¯ s , = ¯ s , . Thus in this twocases the dimension of V is at most 4. Moreover, if (¯ λ , ¯ λ ′ , ¯ λ , ¯ λ ′ ) = ( β , β , β , β ),then from the identity ¯ s , ¯ s , − ¯ s , ¯ s , = 0 we get ¯ s , = − β (¯ a + ¯ a + ¯ a ) andhence the dimension is at most 3.For (¯ λ , ¯ λ ′ , ¯ λ , ¯ λ ′ ) in n (1 , , , , (0 , , , , (cid:16) α , , α , (cid:17)o , from the identity ¯ D = 0 we get ¯ a − = ¯ a . Then, from the identity ¯ d = 0 we deduce¯ a = ¯ a and hence ¯ s , = (1 − β )¯ a . Hence in this cases V has dimension at most 3.Suppose (¯ λ , ¯ λ ′ , ¯ λ , ¯ λ ′ ) = (0 , , , s , ¯ s , + (2 β − a we get ¯ s , = − β (¯ a + ¯ a ) and so ¯ a ¯ a = 0. Hence in this case V is isomorphicto the algebra 2 B . Finally, suppose (¯ λ , ¯ λ ′ , ¯ λ , ¯ λ ′ ) = (1 , , , s , − ¯ s f , = 0 we get ¯ a = ¯ a , that is V is the algebra 1 A .Thus, for each (¯ λ , ¯ λ ′ , ¯ λ , ¯ λ ′ ) in the set (cid:26)(cid:18) β , β , β , β (cid:19) , ( q ( α, β ) , q ( α, β ) , q ( α, β ) , q ( α, β )) , (cid:16) α , , α , (cid:17)(cid:27) , we get that V satisfies the same multiplication table as the algebras 3 C ( β ), 3 C ( α )and 3 A ( α, β ) respectively and has at most the same dimension. Therefore, toconclude the proof, we need only to show that the algebras 3 C ( α ), 3 C ( β ) and3 A ( α, β ) are simple. This follows from Lemma 5.3. (cid:3) As a corollary of Theorem 1.3, we can prove now Theorem 1.6.
Proof of Theorem 1.6.
Let F be a field of characteristic zero. Then F contains Q .The resultant with respect to z of the polynomials in T s has degree 9 and splits in Q [ x ] as the product of a constant and the linear factors x, x − , x − , (cid:18) x − (cid:19) , x − , x − , x − , x − . In Q , the set Z ( T ) consists of the 9 points(1 , , , , (0 , , , , ( 18 , , , , ( 164 , , ,
164 ) , ( 132 , , , ) , ( 132 , , , , ( 164 , , ,
18 ) , ( 32 , , , ) , ( 52 , , , ) . By [11, 7], each quadruple of the above list corresponds to a Norton-Sakuma algebra.By Corollary 4.13 in [9], every Norton-Sakuma algebra is simple, provieded it is notof type 2 B . Hence, the thesis follows from Theorem 1.3, once we prove that in eachcase the dimension of V is at most equal to the dimension of the correspondingNorton-Sakuma algebra. This can be done, by Remark 4.10, using [6]. (cid:3) References [1] Decker, W.; Greuel, G.-M.; Pfister, G.; Sch¨onemann, H.:
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