Triality: from Geometry to Cohomology

M-A. Knus, ETH Zürich

Ramification in Algebra and Geometry at EmoryAtlanta, 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 historyWikipedia:

“There is a geometrical version of triality, analogous to duality in

projective geometry.

... one finds a curious phenomenon involving 1, 2, and 4

dimensional subspaces of 8-dimensional space ...”

Basic object : The algebraic group PGO+8 = PO+

8 /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,

PO8 = {f ∈ GL8 | q(f (x)

)= µ(f )q(x)},

where µ(f ) ∈ F× is the multiplier of f .

Any φ ∈ PO8 acts on C0(q), C0(φ)(xy) = µ−1φ(x)φ(y) and

PGO+8 = {φ ∈ PO8 | C0(φ)|Z = IdZ}, Z = center of C0(q).

Projective GeometryPGO8 is the group of the geometry on the quadric Q6 in P7

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 PGO+8 is the subgroup of PGO8 which respect the

two types.

Projective subspaces ofQ2 of maximal dimension1 in P3

Geometric Triality, Eduard Study (1862-1930)

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

I. The variety of 3-dimensional spaces of a fixed type in Q6 isisomorphic 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 andsolids of the other type are permuted.

Triality, Élie Cartan (1869-1951)

Le principe de dualité et lathéorie des groupes simples etsemi-simples. Bull. SciencesMath. (2) 49 (1925), p. 373 :

“Nous avons donc finalement, à toute substitution portant sur les troisindices 0,1,2, fait correspondre une famille continue de transformationschangeant deux éléments unis en deux éléments unis et deux élémentsen incidence en deux éléments en incidence.L’ensemble de toutes ces transformations forme un groupe mixte, forméde six familles discrètes, qui prolonge le groupe conforme de la mêmemanière que le groupe des homographies et des corrélations prolonge legroupes des homographies en géométrie projective. On peut dire que leprincipe de dualité est remplacé ici par un “principe de trialité”.

In modern language

There is a split exact sequence

1→ PGO+8 → Aut(PGO+

8 )→ S3 → 1

where the homomorphism PGO+8 → Aut(PGO+

8 ) is given by

inner automorphisms

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

8 ).

and S3 occurs as the automorphism group of the Dynkin

diagram of D4 :

dα1

dα2

dα3

dα4

��

TT

Triality and Octonions (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 7→ B, τ : A 7→ C induce σ, τ ∈ Aut(

PGO+(n))

suchthat

σ3 = 1, τ2 = 1, 〈σ, τ〉 = S3 in Aut(

PGO+(n)).

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 7→ Ka 7→ Ra 7→ a.

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 spinors

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 exceptionalgroups

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 1,2D4,⇒ G = PGO+(n),

n norm of some octonion algebra.

II There is a split exact sequence

1→ PGO+(n)→ Aut(

PGO+(n))→ S3 → 1

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. PGO+8 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 composition algebra 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 trialityTheorem Let (S, ?, n) be a symmetric composition of dimension 8and let f ∈ GO+(n). There exists f1, f2 ∈ GO+(n) such that

µ(f )−1f (x ? y) = f2(x) ? f1(y)µ(f1)−1f1(x ? y) = f (x) ? f2(y)µ(f2)−1f2(x ? y) = f1(x) ? f (y).

One formula implies the two others. The pair (f1, f2) is uniquelydetermined by f up to a scalar (λ, λ−1).

Proof Hint ([KMRT]) : Use x ? (y ? x) = (x ? y) ? x = n(x)y to showthat 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

ρ2?([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] 7→ [a ? S] 7→ [S ? a] 7→ [a], a ∈ S

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

trialitarian automorphism ρ differs from ρ? (or ρ2?) by an inner

automorphism, let

ρ = Int([f ]

)◦ ρ?

then ρ = ρ� with x � y = f f−11

[f f1f−1(x) ? f−1(y)

].

[Recall µ(f )−1f (x ? y) = f2(x) ? f1(y)].

Remark ρ3 = 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}.

x ? y =xy − ωyx

1− ω − 13

tr(xy), n(x) = −16

Trd(x2).

(A0, ?, n) is a symmetric composition whose norm is of

maximal index.

II AutF (A0, ?) = PGL1(A).

The split case was discovered by thetheoretical physicist Sumusu Okuboen 1964.

Let ω 6∈ F .

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×.

II AutF (A0, ?) = PGU1(B, τ).

Okubo algebras

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

Okubo symmetric compositions.

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

Any symmetric composition is of type G2 or A2

Question : Fields of characteristic 3 ?

A theory of independent interest !

Groups with triality of outer type 3,6D4

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

H1(F ,PGO+8 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 PGO+8 oS3 (Parimala, Sridharan, K., 2000).

III. David’s contribution to triality

Triality, Cocycles, Crossed Products, Involutions, Cli!ord Algebras and Invariants

David J. Saltman!

Department of MathematicsThe 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 = PO+

8 !| S3. Weshow how this can be translated to the invariants of the trialitarian Weylgroup (S2)3!| S4)!| S3 and then work out the consequences. As it turnsout, these consequences lead one to a whole host of subjects. Thus inthe paper we define so called G " H cocycles, and the associated Azu-maya crossed products. We define well situated involutions and describethem on the above named crossed products, associating them with cer-tain “splittings”. We also describe a group of algebras with well situatedinvolutions that maps to the Brauer group. We define Cli!ord algebrasfor Azumaya algebras with involution, and show this map has the formof a fixed element plus a homomorphism when restricted to well situatedinvolutions and a fixed splitting subring. With all this work we can thengive 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, Cli!ordalgebras, G"H cocycles, Crossed products, Azumaya algebras

*The author is grateful for support under NSF grant DMS-9970213. The authorwould also like to acknowledge the hospitality of the Universite Catolique de Louvainduring part of the preparation of this paper.

Reduction to the Weyl group

Fix a split torus T ⊂ PGO+8 invariant under a trialitarian action

to get an induced trialitarian action on the Weyl group

W = S32 o S4 of PGO+

8 .

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 ,S32 o S4) ⇔ 8-dim. étale algebras with inv. and triv. disc.

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

(A, σ) is well situated with respect to (L, σ′)⇔ L ⊂ A and σL = σ′.

Classical invariants, like Clifford invariant and discriminant extend forfor well situated involutions.

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

End