Triality: from Geometry to Cohomology

M-A. Knus, ETH Zürich

Ramification in Algebra and Geometry at Emory Atlanta, May 18, 2011 Outline

I Introduction and some historical comments

I A different approach at triality

I David’s work on triality I. Introduction and some history

Wikipedia:

“There is a geometrical version of triality, analogous to in projective geometry.

... one finds a curious phenomenon involving 1, 2, and 4 dimensional subspaces of 8-dimensional space ...” + + Basic object : The algebraic group PGO8 = PO8 /F × Let

I F a field of characteristic not 2 and 3,

I q a quadratic form of dimension 8 over F and of maximal index,

I PO8 the group of similitudes of q,

PO = f GL qf (x) = µ(f )q(x) , 8 { ∈ 8 | } where µ(f ) F × is the multiplier of f . ∈

Any φ PO acts on C (q), C (φ)(xy) = µ−1φ(x)φ(y) and ∈ 8 0 0 PGO+ = φ PO C (φ) = Id , Z = center of C (q). 8 { ∈ 8 | 0 |Z Z } 0 Projective Geometry 6 7 PGO8 is the group of the geometry on the quadric Q in P defined by q = 0. There are two types of projective subspaces of Q6 of maximal dimension 3:

*solids of type I and solids of type II*.

+ The group PGO8 is the subgroup of PGO8 which respect the two types.

Projective subspaces of Q2 of maximal dimension 1 in P3 Geometric Triality, Eduard Study (1862-1930)

Grundlagen und Ziele der analytischen Kinematik, Sitzungsberichte der Berl. Math. Gesell. (1913), p. 55 :

I. The variety of 3-dimensional spaces of a fixed type in Q6 is isomorphic to a quadric Q6. II. Any proposition in the geometry of Q6 [about incidence rela- tions] remains true if the concepts points, solids of one type and solids of the other type are permuted. Triality, Élie Cartan (1869-1951)

Le principe de dualité et la théorie des groupes simples et semi-simples. Bull. Sciences Math. (2) 49 (1925), p. 373 :

“Nous avons donc finalement, à toute substitution portant sur les trois indices 0, 1, 2, fait correspondre une famille continue de transformations changeant deux éléments unis en deux éléments unis et deux éléments en incidence en deux éléments en incidence. L’ensemble de toutes ces transformations forme un groupe mixte, formé de six familles discrètes, qui prolonge le groupe conforme de la même manière que le groupe des homographies et des corrélations prolonge le groupes des homographies en géométrie projective. On peut dire que le principe de dualité est remplacé ici par un “principe de trialité”. In modern language

There is a split exact sequence

1 PGO+ Aut(PGO+) S 1 → 8 → 8 → 3 → where the homomorphism PGO+ Aut(PGO+) is given by 8 → 8 inner automorphisms

g PGO+ Int(g)(x) = gxg−1 Aut(PGO+). ∈ 8 7→ ∈ 8

and S3 occurs as the of the Dynkin

diagram of D4 : α3   d α1 α2 TT d d α4 d Triality and (Cartan,1925)

Let O be the 8-dimensional algebra of Cayley numbers (octonions) and let n be its norm.

Given A SO(n) there exist B, C SO(n) such that ∈ ∈ C(x y) = Ax By. · · σ : A B, τ : A C induce σ, τ Aut PGO+(n) such that 7→ 7→ ∈

σ3 = 1, τ 2 = 1, σ, τ = S in Aut PGO+(n). h i 3 A swiss PhD-Student of É. Cartan

Félix Vaney, Professeur au Collège cantonal, Lausanne (1929) :

I Solids are of the form

I. Ka = x O ax = 0 and II. Ra = x O xa = 0 . { ∈ | } { ∈ | } II Geometric triality can be described as

a Ka Ra a. 7→ 7→ 7→

for all a O with n(a) = 0. ∈ A selection of later works

E. A. Weiss (1938,1939) : More (classical) projective geometry É. Cartan (1938) : Leçons sur la théorie des spineurs N. Kuiper (1950) : Complex algebraic geometry H. Freudenthal (1951) : Local and global triality C. Chevalley (1954) : The algebraic theory of J. Tits (1958) : Triality for loops J. Tits (1959) : Classification of geometric trialities over arbitrary fields F. van der Blij, T. A. Springer (1960) : Octaves and triality T. A. Springer (1963) : Octonions, Jordan algebras and exceptional groups N. Jacobson (1964) : Triality for Lie algebras over arbitrary fields.

Books (Porteous, Lounesto, [KMRT], Springer-Veldkamp). Classification of trialitarian automorphisms

G simple adjoint group of type D4 with trialitarian action.

I G of type 1,2D , G = PGO+(n), 4 ⇒ n norm of some algebra.

II There is a split exact sequence

1 PGO+(n) Aut PGO+(n) S 1 → → → 3 → Thus there is up to inner automorphisms essentially one trialitarian automorphism of PGO+(n). Triality up to conjugacy

Of independent interest : classification of trialitarian automorphisms up to conjugacy in the group of automorphisms.

1. Geometric trialities with absolute points (Tits,1959) 2. Simple Lie algebras over algebraically closed fields (Wolf-Gray,1968, Kac, 1969,1985...) 3. Simple orthogonal Lie algebras over fields of characteristic 0 (K., 2009) + 4. PGO8 over finite fields (Gorenstein, Lyons, Solomon,1983) 5. Simple algebraic groups of classical type over arbitrary fields (Chernousov, Tignol, K., 2011) II. A different approach to triality

Markus Rost ( 1991) : There is a class of composition ∼ algebras well suited for triality !

“Symmetric compositions”

See [KMRT]. Symmetric compositions

A is a quadratic space (S, n) with a bilinear multiplication ? such that the norm of multiplicative :

n(x ? y) = n(x) ? n(y)

They exist only in dimension 1, 2, 4 and 8 (Hurwitz).

A symmetric composition satisfies

x ? (y ? x) = (x ? y) ? x = n(x)y and bn(x ? y, z) = bn(x, y ? z)

Remark Already studied in a different setting (Petersson, Okubo, Elduque, ...) ! Symmetric compositions and triality Theorem Let (S, ?, n) be a symmetric composition of dimension 8 + + and let f GO (n). There exists f1, f2 GO (n) such that ∈ ∈ 1 µ(f )− f (x ? y) = f2(x) ? f1(y) 1 µ(f1)− f1(x ? y) = f (x) ? f2(y) 1 µ(f2)− f2(x ? y) = f1(x) ? f (y).

One formula implies the two others. The pair (f1, f2) is uniquely 1 determined by f up to a scalar (λ, λ− ). Proof Hint ([KMRT]) : Use x ? (y ? x) = (x ? y) ? x = n(x)y to show  that C(q) = M2 EndF S . + + Corollary The map ρ? : PGO (n) PGO (n) defined by → + ρ?([f ]) = [f2], f GO (n) ∈ + is a trialitarian automorphism of PGO (n). Observe that ρ? satisfies

2 ρ?([f ]) = [f1]. Symmetric compositions and projective geometry

Let (S, ?, n) be a symmetric composition and let Q6 = n(x) = o . { }

I The volumes of one type on Q 6 are of the form [a ? S] and of the form [S ? a], a with n(a) = 0, for the other type. II The permutation [a] [a ? S] [S ? a] [a], a S 7→ 7→ 7→ ∈ is a geometric triality.

Remark The idea to characterize isotropic spaces through symmetric compositions came out of a note of Eli Matzri. Classification of symmetric compositions and trialities

Theorem (Chernousov, Tignol, K., 2011):

Isomorphism classes of symmetric compositions with norm n

⇔ Conjugacy classes of trialitarian automorphisms of PGO+(n) One step in the proof

Symmetric compositions induce trialitarian automorphisms. Conversely, any trialitarian automorphism is induced by a symmetric composition:

Fix (S, ?, n) and let ρ? be induced by (S, ?, n). Any other 2 trialitarian automorphism ρ differs from ρ? (or ρ?) by an inner automorphism, let  ρ = Int [f ] ρ? ◦ −1 −1 −1  then ρ = ρ with x y = f f f f f (x) ? f (y) .  1 1 −1 [Recall µ(f ) f (x ? y) = f2(x) ? f1(y)].

3 Remark ρ = 1 implies f f1f2 = IdS. Consequences

1. The classification of symmetric compositions (Elduque-Myung, 1993) yields the classification of conjugacy classes of trialitarian automorphisms of groups PGO+(n).

2. Conversely one can first classify conjugacy classes of trialitarian automorphisms of groups PGO+(n) (Chernousov, Tignol, K., 2011) and deduce from it the classification of symmetric compositions. Symmetric compositions of type G2

An octonion algebra is not a symmetric composition ! However : I The norm n of a symmetric composition is a 3-Pfister form, i.e. the norm of a octonion algebra.

II If (O, , n) is an octonion algebra, then ·

x ? y = x y, x, y O, · ∈ defines a symmetric composition, called para-octonion.

III Aut(S, ?, n) an algebraic group of type G2. Conclusion : For any such n there is at least one symmetric composition ! Symmetric compositions of type A2 Let ω F, ω3 = 1. ∈ I A central simple of degree 3 over F, A0 = x S Trd(x) = 0 . { ∈ | } xy ωyx 1 1 x ? y = − tr(xy), n(x) = Trd(x2). 1 ω − 3 −6 − (A0, ?, n) is a symmetric composition whose norm is of maximal index.

0 II AutF (A ,?) = PGL1(A).

The split case was discovered by the theoretical physicist Sumusu Okubo en 1964. Let ω F. 6∈ I (B, τ) central simple of degree 3 over F(ω) with unitary involution τ, B0 = x B τ(x) = x, Trd(x) = 0 , ? and n { ∈ | } as above. Then (B0, ?, n) is a symmetric composition whose norm is is a 3-Pfister form 3, α, β , for some α, β F ×. hh ii ∈

0 II AutF (A ,?) = PGU1(B, τ). Okubo algebras

(A0, ?, n) and (B0, ?, n) are now called

Okubo symmetric compositions.

1,2 Their automorphism groups are algebraic groups of type A2.

Any symmetric composition is of type G2 or A2

Question : Fields of characteristic 3 ?

A theory of independent interest ! 3,6 Groups with triality of outer type D4

Outer types are related with

I Semilinear trialities (in projective geometry)

I Generalized hexagons (incidence geometry, Tits, Schellekens, ...)

I A simple group (Tits, Steinberg, Hertzig)

I Twisted compositions (F4, Springer)

I Trialitarian algebras (KMRT) Trialitarian algebras

Trialitarian algebras are *algebras* classified by 1 + H (F, PGO8 oS3). They consist of 1. A cubic étale algebra L/F, 2. A central simple algebra of degree 8 with orthogonal involution (A, σ) over L, 3. More conditions ... (see [KMRT])

There is a *generic trialitarian algebra* whose center is a field + of invariants of PGO8 oS3 (Parimala, Sridharan, K., 2000). III. David’s contribution to triality

Triality, Cocycles, Crossed Products, Involutions, Clifford Algebras and Invariants

David J. Saltman∗ Department of Mathematics The University of Texas Austin, Texas 78712

Abstract In [KPS] a “generic” trialitarian algebra was defined and de- + scribed using the invariants of the trialitarian group T = PO8 | S3. We show how this can be translated to the invariants of the trialitarian× Weyl 3 group (S2) | S4) | S3 and then work out the consequences. As it turns out, these consequences× × lead one to a whole host of subjects. Thus in the paper we define so called G H cocycles, and the associated Azu- maya crossed products. We define− well situated involutions and describe them on the above named crossed products, associating them with cer- tain “splittings”. We also describe a group of algebras with well situated involutions that maps to the Brauer group. We define Clifford algebras for Azumaya algebras with involution, and show this map has the form of a fixed element plus a homomorphism when restricted to well situated involutions and a fixed splitting subring. With all this work we can then give an intrinsic description of trialitarian algebras as in Theorem 9.15.

AMS Subject Classification: 20G15, 16W10, 16H05, 12E15, 16K20, 12G05, 14F22, 16S35, 11E04, 11E88, 13A50, 14L30 Key Words: triality, trialitarian group, well situated, involutions, Clifford algebras, G H cocycles, Crossed products, Azumaya algebras −

*The author is grateful for support under NSF grant DMS-9970213. The author would also like to acknowledge the hospitality of the Universite Catolique de Louvain during part of the preparation of this paper. Reduction to the Weyl group

Fix a split torus T PGO+ invariant under a trialitarian action ⊂ 8 to get an induced trialitarian action on the Weyl group 3 + W = S2 o S4 of PGO8 . Aim of David (among others) : Use triality on W to give an intrinsic description of trialitarian algebras. For this constructions and techniques were needed which are useful in many other subjects.

Examples New kinds of cocycles, crossed products, Clifford algebras, well situated involutions, .... Well situated involutions

H1(F, S3 S ) 8-dim. étale algebras with inv. and triv. disc. 2 o 4 ⇔

H1(F, PGO+) c.s. alg. of deg. 8 with orth. inv. and triv. disc. 8 ⇔

(A, σ) is well situated with respect to (L, σ0) L A and σL = σ0. ⇔ ⊂ Classical invariants, like Clifford invariant and discriminant extend for for well situated involutions.

Severi-Brauer varieties for well situated involutions (Tignol, K., 2011) End