aa r X i v : . [ m a t h . AG ] F e b MOUFANG LOOPS AND TORIC SURFACES
Picard-Lefschetz reflections in the nonassociative world
Vadim SchechtmanFebruary 26, 2021 §0. Introduction.
It was establshed in the early 1930-th by the mathematicians of the Hilbert school(Emil Artin, Max Zorn, Ruth Moufang) that the notion of a "field"(
K¨orper ) K is thesame as the notion of a projective plane P ( K ) : the laws of addition and multiplicationin K may be restored geometrically from the incidence relations of points and lines in P ( K ) . The associativity of multiplication is equivalent to the Desargue theorem, sothat a plane where this theorem is true corresponds to a skew-field ( Schiefk¨orper ).R.Moufang considered a plane with a weakened form of the Desargue theorem - theso called der Satz vom volst¨andigen Vierseit ( D ) ; the set of nonzero elements of thecorresponding field ( Alternativk¨orper ) carries a nonassociative structure called today a
Moufang loop .In this note we propose a construction of some varieties which might be considered as"nonsingular toric surfaces" X L over a Moufang loop L . Suppose that L = K \{ } where K is a Cayley octonion division algebra over a commutative field k . Then, as is usual inthe toric geometry, X L is patched together out of several copies of affine planes K . Thetransition functions between various charts have a nice triangular "noncommutativePicard-Lefschetz" shape. This way we get certain -dimensional varieties over k .We restrict ourselves to some examples. The general case depends on some combinatorial"lifting conjecture" , see 4.4 below, which might be of interest in itself. We note thatthe gluing construction does not use the full strength of the Moufang identity but onlya weaker inversion property, see 1.1 below.I am grateful to B.Toen and J.Tapia for inspiring discussions.1
1. Moufang loops1.1. Loops.
Recall that a magma (in the terminology of N.Bourbaki) is a set L equipped with an operation L × L −→ L, ( x, y ) xy (1 . . which need not to be neither associative nor commutative.Let us call a loop a magma L equipped with a distingushed element and a bijection L ∼ −→ L, x x − such that for all x, y ∈ L (i) x x = 1 x ;(ii) ( x − ) − = 1; x − x = xx − = 1; (iii) x − ( xy ) = ( yx ) x − = y, cf. [CS] 7.1 .It is shewn in op. cit. L ( yx ) y = y ( xy ) (1 . . for all x, y ; we denote this product yxy . It follows that for a ∈ Z ≥ is defined in x a unambiguous manner by induction: x = 1 , x a = x ( x a − ) = ( x a − ) x ; (1 . . we define x − a := ( x a ) − . In a loop for all x, y ( xy ) − = y − x − . (1 . . where it is called a loop with inversion . roof [H, Lemma 2.11] ( xy ) − = y − ( y ( xy ) − ) = y − (( x − ( xy ))( xy ) − ) = y − x − . (cid:3) Note that in this proof the unity has not been used. Recall that a
Moufang loop is a loop L satisfying the axiom:— for all x, y, z z ( xy ) z = zx · zy, ( a ) z ( xy ) = ( zxz )( z − y ) , ( b )( xy ) z = ( z − )( zyz ); ( c ) in fact any of these identities implies two others, cf. [CS], 7.4. (see [Z], Satz 4, p. 127, [M2]) A Moufang loop with two generators is associative. (cid:3) §2. Cayley-Moufang projective plane2.1. Projective line and projective plane in the toric language.
Let k be afield.(a) The projective line P ( k ) with coordinates ( t : t ) is covered by two opens bothisomorphic to an affine line A ( k ) = k where we introduce the coordinates U = { t = 0 } , x := t /t and U = { t = 0 } , x ′ := t /t ; their intersection admits two coordinates U = U ∩ U ∼ = G m ( k ) = k ∗ , x ′ = x − . (2 . . So P ( k ) is obtained by gluing of two copies of A ( k ) along G m ( k ) , with a transitionfunction given by (2.1.1). 3b) The projective plane P ( k ) with coordinates ( t : t : t ) is covered by threeopens U i = { t i = 0 } isomorphic to an affine plane A ( k ) = k .We mark the coordinates in these charts:in U : x = t /t , y = t /t ;in U : x ′ = t /t , y ′ = t /t ;in U : x ′′ = t /t , y ′′ = t /t .Double intersections U ij ∼ = k ∗ × k ; transition functions φ ( x, y ) = ( x ′ , y ′ ) , x ′ = x − , y ′ = x − y (2 . . a ) on U = { ( x, y ) ∈ U | x = 0 } = { ( x ′ , y ′ ) ∈ U | x ′ = 0 } ; φ ( x ′ , y ′ ) = ( x ′′ , y ′′ ) , x ′′ = y ′− x ′ , y ′′ = y ′− , (2 . . b ) on U = { ( x ′ , y ′ ) ∈ U | y ′ = 0 } = { ( x ′′ , y ′′ ) ∈ U | y ′′ = 0 } ; φ ( x, y ) = ( x ′′ , y ′′ ) , x ′′ = y − , y ′′ = y − x. (2 . . c ) on U = { ( x, y ) ∈ U | y = 0 } = { ( x ′′ , y ′′ ) ∈ U | x ′′ = 0 } . On the triple intersection U ∼ = k ∗ we have three coordinate systems: ( x, y ) , ( x ′ , y ′ ) and ( x ′′ , y ′′ ) . On the triple intersection U we have φ = φ φ . (2 . . Proof.
We have x ′′ = y ′− x ′ = ( y − x ) x − = y − and y ′′ = y ′′ = y ′− = ( x − y ) − = y − x. (cid:3) Note that in this computation we have only used the identities 1.1 (i) - (iii), nocommutativity and a weak form of the associativity (1.1.4).
Let L be a loop. We denote an affine line for L A = A ( L ) = L a { } .
4e define a projective line P ( L ) as the quotient of a disjoint union A ( L ) × A ∞ ( L ) where A ∞ ( L ) = L ` {∞} under an identification x ′ = x − where x ∈ L ⊂ A ( L ) (resp. x ′ ∈ L ⊂ A ∞ ( L ) ). Thus P ( L ) = { } a L a {∞} . We define the projective plane P ( L ) as a quotient of a disjoint union of three copies A a A a A where A (resp. A , A ) is a copy of A ( L ) = A × A with coordinates ( x, y ) (resp. ( x ′ , y ′ ) , ( x ′′ , y ′′ ) ) under an equivalence relation: ( x ′ , y ′ ) = φ ( x, y ) , x = 0 , ( x ′′ , y ′′ ) = φ ( x ′ , y ′ ) , y ′ = 0 , ( x ′′ , y ′′ ) = φ ( x, y ) , y = 0 . Here φ : L × A ∼ −→ L × A,φ : A × L ∼ −→ A × L,φ : A × L ∼ −→ L × A. This set is well defined due to the key
We have φ = φ φ on the triple intersection L × L . Proof
The computation in the proof of 2.1.1 uses only the identities 1.1 (i) - (iii),no commutativity and a weak form of the associativity. (cid:3)
We have by definition P ( L ) = C a C a C C = U ∼ = A ,C = U \ U ∼ = A,C = U \ ( U ∪ U ) ∼ = {∗} , this is what Tits calls a coordinate system on the Cayley projective plane, cf. [T],9.11.2. Let L be the variety of nonzero elements of the real Cayleyoctonions. Then P L = S , and P L is a -dimensional real variety - the Cayley projectiveplane. §3. Octonionic toric surfaces3.1. Hirzebruch surfaces. (a) Commutative version
Let k be a field, a > be an integer. Following [F], Introduction, p.7, a Hirzebruchsurface F a is defined by patching four affine charts U i ∼ = k , ≤ i ≤ with coordinatesand transition functions: ( x, y ) on U , ( x ′ , y ′ ) = φ ( x, y ) = ( y − , x ) on U , ( x ′′ , y ′′ ) = φ ( x, y ) = ( x − , x − a y − ) on U , ( x ′′′ , y ′′′ ) = φ ( x, y ) = ( yx a , x − ) on U .(b) A loop version
Let L be a loop. We define A = A ( L ) as above in §2, and consider four copies U i ∼ = A × A with coordinates ( x, y ) , ( x ′ , y ′ ) , etc.; inside them six subsets U = A × L ⊂ U , U = L × A ⊂ U ; U = A × L ⊂ U , U = L × A ⊂ U ; U = A × L ⊂ U , U = L × A ⊂ U ; next U = L × L ⊂ U , U = L × L ⊂ U ; = L × L ⊂ U , U = L × L ⊂ U . next U = L × A ⊂ U , U = A × L ⊂ U . Define six patching functions (recall that x ± a is defined in (1.1.3)): φ : U ∼ −→ U , φ ( x, y ) = ( y − , x ) ,φ : U ∼ −→ U , φ ( x ′ , y ′ ) = ( y ′− , y ′− a x ′ ) ,φ : U ∼ −→ U , φ ( x ′′ , y ′′ ) = ( y ′′− , x ′′ ) , and φ : U ∼ −→ U , φ ( x, y ) = ( x − , x − a y − ) ,φ : U ∼ −→ U , φ ( x ′ , y ′ ) = ( x ′− y ′ a , y ′− ) , and finally φ : U ∼ −→ U , φ ( x, y ) = ( yx a , x − ) . For all ≤ i < j < k ≤ φ ik = φ jk φ ij . One has to check four relations corresponding to ( ijk ) = (012) , (013) , (023) and (123) .This is done directly. P blown up at one point. Classically this variety which we denote X maybe obtained as the toric variety associated with a fan with four generators v , . . . , v ∈ Z where v = (1 , , v = (1 , , v = (0 , , v = ( − , − , we number them in the clockwise order, cf. [F], 2.4. Thus X = ∪ i ∈ Z / Z U i where U i is the chart ∼ = k corresponding to the dual of the cone with generators v i , v i +1 .7et us describe the corresponding Moufang variety X ( L ) where L is a loop. It ispatched from four charts U i ( L ) ∼ = A ( L ) whose coordinates we denote by ( u, v ) , ( u ′ , v ′ ) , etc. It is convenient to express them in terms of coordinates x, y usual in toric geometry.The formulas below should be considered as a definition of certain couples ofelements in the free inversion loop on generators x, y ; they are the usual toric formulasbut the order of factors is important. ( u, v ) = ( x, x − y ) , ( u ′ , v ′ ) = ( y − x, y )( u ′′ , v ′′ ) = ( y − , y − x ) , ( u ′′′ , v ′′′ ) = ( x − y, x − ) (3 . . Returning to our four charts, we define inside them the subsets correspondingto double intersections U ij ⊂ U i , i = j , with U ∼ = A × L, U ∼ = L × A,U ∼ = A × L, U ∼ = L × A,U ∼ = A × L, U ∼ = L × A,U ∼ = U = L × LU ∼ = U ∼ = L × L,U ∼ = L × A, U ∼ = A × L. We define six patching functions: φ : U ∼ −→ U , φ ( u, v ) = ( u ′ , v ′ ) = ( v − , uv ); φ : U ∼ −→ U , φ ( u ′ , v ′ ) = ( u ′′ , v ′′ ) = ( v ′− , u ′ ); φ : U ∼ −→ U , φ ( u ′′ , v ′′ ) = ( u ′′′ , v ′′′ ) = ( v ′′− , v ′′− u ′′ ); next φ : U ∼ −→ U , φ ( u, v ) = ( u ′′ , v ′′ ) = ( v − u − , v − ); φ : U ∼ −→ U , φ ( u ′ , v ′ ) = ( u ′′′ , v ′′′ ) = ( u ′− , u ′− v ′− ); finally φ : U ∼ −→ U , φ ( u, v ) = ( u ′′′ , v ′′′ ) = ( v, u − ) .
8e need to check four transitivity relations: φ ik = φ jk φ ij for ( i, j, k ) = (0 , , , (0 , , , (0 , , , or (1 , , . This is done readily. One uses onlythe axiom 1.1 (iii), (1.1.4). A general nonsingular complete toric surface X is ablowing up at several points either of F a or of P , cf. [F], 2.5.Their loop versions are discussed in the next Section. §4. Nice cyclic collections4.1. Elementary matrices and pseudoreflections. Let a ∈ Z . We consider fourkinds of matrices, to be called elementary , belonging to SL ( Z ) : A ( a ) = (cid:18) − a (cid:19) , A ′ ( a ) = (cid:18) a − (cid:19) ,B ( a ) = (cid:18) − − a − (cid:19) , B ′ ( a ) = (cid:18) − a − (cid:19) The inverse to an elementary matrix is elementary: A ′ ( a ) = A ( a ) − , B ( a ) − = B ( − a ) , B ′ ( a ) − = B ′ ( − a ) (4 . . but the product of elementary matrices of different kinds is not elementary.To each elementary matrix X ( a ) (for X ∈ { A, A ′ , B, B ′ } ) we associate two invertibleuniversal transformations X ǫ ( a ) : L × L −→ L × L, ǫ = ± (4 . . defined for every loop L .Namely: A + ( a )( x, y ) = ( y − , xy a ) , A − ( a )( x, y ) = ( y − , y a x ) ,A ′− ( a )( x, y ) = ( x a y, x − ) , A ′− ( a )( x, y ) = ( yx a , x − ) ,B + ( a )( x, y ) = ( x − , x a y − ) , B − ( a )( x, y ) = ( x − , y − x a ) ,B ′− ( a )( x, y ) = ( x − y a , y − ) , B ′− ( a )( x, y ) = ( y a x − , y − ) . A ± ( a ) − = A ′∓ ( a ) , B ± ( a ) − = B ∓ ( − a ) , B ′± ( a ) − = B ′∓ ( − a ) We will say that X ǫ ( a ) is a pseudo-reflection with matrix X ( a ) , or that X ǫ ( a ) is a liftingof X ( a ) .We have seen that the inverse to a pseudoreflection is a pseudoreflection.However the composition of two pseudoreflections is not necessarily a pseudoreflection. Let n ≥ . Let us call a sequence of matrices A = { A ( a i ) , a i ∈ Z , ≤ i ≤ n − } a Fulton toric cycle if A ( a ) A ( a ) . . . A ( a n − ) = I. (4 . . Such a sequence is exactly the data defining a complete nonsingular toric surface, cf.[F], 2.5.Note that (4.2.1) implies A ( a i ) A ( a i +1 ) . . . A ( a n ) A ( a ) . . . A ( a i − ) = I. for any ≤ i ≤ n − .Let us identify the set { , . . . , n − } with Z /n Z in the obvious manner whichprovides us with a cyclic order on this set (we imagine its elements as points p i a circlein the clockwise order). For each i, j ∈ Z /nZ , we define a matrix A ij ( a ) = A ( a j − ) A ( a j − ) . . . A ( a i ) , A ii ( a ) = I (4 . . One should imagine A ij ( a ) as an arrow on the circle A ij ( a ) : p i −→ p j which joins clockwise p i with p j .It seems that the following is true. All matrices A ij ( a ) are elementary. A nice cycle of length n is a collection of pseudoreflections M = { M ij , i, j ∈ Z /n Z } , ij = X ǫ ij ( a ij ) , X = { A, A ′ , B, B ′ } , such that(a) For all i M ii = Id, and M i,i +1 is of type A , i.e. M i,i +1 = A ǫ i ( a i ) , a i ∈ Z , ǫ i = ± . (b) For all i ≤ j ≤ k (in the cyclic order) M ik = M jk M ij . Let us denote M i := M i,i +1 , and a = ( a , . . . , a n − ) ∈ Z Z /n Z . It follows from (b) that M M . . . M n − = Id (4 . . where the order of brackets in the LHS is inessential - the result does not depend onit. More generally M ij = M j − M j − . . . M i , (4 . . for all i, j , the order of brackets in the RHS being inessential.The collection of matrices A ( a ) = { A ( a i ) , i ∈ Z /n Z } is a Fulton cycle. We will say that M is a lifting of A ( a ) . n = 4 ). We have A + ( a ) A (0) A − ( − a ) A (0) = IdHere A (0) := A + (0) = A − (0) . We have: M = A − ( − a ) A (0) = B + ( − a ) , M = A ′− ( a ) , M = B ′ + ( a ) , etc. See 3.1. For every Fulton cycle A ( a ) there exists a lifting M . .5. Given a lifting M we can proceed as in §3, by gluing the corresponding toricsurface X ( M ) from n copies U , . . . , U n of A and using matrices M ij as patchingfunctions M ij : U ij ∼ −→ U ji . Note that if the initial loop L is a Moufang loop we can proceedas follows to define a lifting. Given a Fulton cycle A ( a ) , we start from transformations M i = A ǫ i ( a i ) , ≤ i ≤ n − with arbitrary ǫ i = ± . Then define M ij for ≤ i < j ≤ n − by formula (4.3.2): the position of brackets is inessential due to the diassotiativity 1.4.The cocycle condition 4.3 (b) will be satisfied for ≤ i ≤ j ≤ k ≤ n − . Thesedata is enough to glue a variety. We can define M n − by (4.3.1). The price we payfor an arbitrary choice of ǫ i is that the transformations M ij are in general no longerpseudoreflections. We may call the collection M = { M ij } a mock lifting of A ( a ) . Sothere are n − mock liftings. References [A] E.Artin, Coordinates in affine geometry,
Rep. Math. Coll. Notre Dame (Indiana)
2¯ (1940), 15-20.[B] J.Baez, The octonions,
Bull. AMS (2001), 145-205.[Br] R.H.Bruck, A survey of binary systems, Ergebnisse
Bd. , Springer 1971.[CS] J.H.Conway, D.A.Smith, On quaternions and octonions: their geometry, arithmeticand symmetry, A.K.Peters , 2003.[F] W.Foulton, Introduction to toric varieties,
Ann. Math. Studies , PUP , 1993.[H] J.I.Hall, Moufang loops and groups with triality are essentially the same thing,
Mem. AMS , 2019.[M1] R.Moufang, Alternativk¨orper and der Satz vom volst¨andigen Vierseit ( D ) , Abh. Math. Sem. Univ. Hamburg (1933), 207-222.[M2] R.Moufang, Zur Struktur von Alternativk¨orper, Math. Ann. (1935), 416- 430.[ST] T.A.Springer, F.D.Veldkampf, Octonions, Jordan algebras and exceptionalgroups,
Springer , 2000.[T] J.Tits, Buildings of spherical type and finite BN-pairs,
Lecture Notes in Math. , Springer , 1974.[Z] M.Zorn, Theorie der alternativen Ringe,
Abh. math. Seminar Univ. Hamburg8