Triality: from Geometry to Cohomology M-A. Knus, ETH Zrich - - PowerPoint PPT Presentation
Triality: from Geometry to Cohomology M-A. Knus, ETH Zrich Ramification in Algebra and Geometry at Emory Atlanta, May 18, 2011 Outline Introduction and some historical comments A different approach at triality Davids work on
M-A. Knus, ETH Zürich
Ramification in Algebra and Geometry at Emory Atlanta, May 18, 2011
◮ Introduction and some historical comments ◮ A different approach at triality ◮ David’s work on triality
Wikipedia: “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 ...”
8 = PO+ 8 /F ×
Let
◮ F a field of characteristic not 2 and 3, ◮ q a quadratic form of dimension 8 over F and of maximal
index,
◮ PO8 the group of similitudes of q,
PO8 = {f ∈ GL8 | q
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).
PGO8 is the group of the geometry on the quadric Q6 in P7 defined by q = 0. There are two types of projective subspaces
*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 of Q2 of maximal dimension 1 in P3
Grundlagen und Ziele der analytischen Kinematik, Sitzungsberichte der Berl. Math.
isomorphic to a quadric Q6.
tions] remains true if the concepts points, solids of one type and solids of the other type are permuted.
Le principe de dualité et la théorie des groupes simples et semi-simples. Bull. Sciences
“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”.
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 → Int(g)(x) = gxg−1 ∈ Aut(PGO+ 8 ).
and S3 occurs as the automorphism group of the Dynkin diagram of D4 :
❞
α1
❞
α2
❞α3 ❞α4 ✔ ✔ ❚ ❚
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
that σ3 = 1, τ 2 = 1, σ, τ = S3 in Aut
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. for all a ∈ O with n(a) = 0.
É. Cartan (1938) : Leçons sur la théorie des spineurs
F . van der Blij, T. A. Springer (1960) : Octaves and triality
groups
Books (Porteous, Lounesto, [KMRT], Springer-Veldkamp).
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
Thus there is up to inner automorphisms essentially one trialitarian automorphism of PGO+(n).
Of independent interest : classification of trialitarian automorphisms up to conjugacy in the group of automorphisms.
(Wolf-Gray,1968, Kac, 1969,1985...)
characteristic 0 (K., 2009)
8 over finite fields (Gorenstein, Lyons, Solomon,1983)
fields (Chernousov, Tignol, K., 2011)
Markus Rost (∼1991) : There is a class of composition algebras well suited for triality ! “Symmetric compositions” See [KMRT].
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, ...) !
Theorem Let (S, ⋆, n) be a symmetric composition of dimension 8 and 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 uniquely determined by f up to a scalar (λ, λ−1). Proof Hint ([KMRT]) : Use x ⋆ (y ⋆ x) = (x ⋆ y) ⋆ x = n(x)y to show that C(q) = M2
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].
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
II The permutation [a] → [a ⋆ S] → [S ⋆ a] → [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.
Theorem (Chernousov, Tignol, K., 2011): Isomorphism classes of symmetric compositions with norm n ⇔ Conjugacy classes of trialitarian automorphisms of PGO+(n)
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
then ρ = ρ⋄ with x ⋄ y = f f −1
1
[Recall µ(f)−1f(x ⋆ y) = f2(x) ⋆ f1(y)]. Remark ρ3 = 1 implies f f1f2 = IdS.
(Elduque-Myung, 1993) yields the classification of conjugacy classes of trialitarian automorphisms of groups PGO+(n).
trialitarian automorphisms of groups PGO+(n) (Chernousov, Tignol, K., 2011) and deduce from it the classification of symmetric compositions.
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 !
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 − 1 3 tr(xy), n(x) = −1 6 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 the theoretical physicist Sumusu Okubo en 1964.
Let ω ∈ 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).
(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 !
Outer types are related with
◮ Semilinear trialities (in projective geometry) ◮ Generalized hexagons (incidence geometry, Tits,
Schellekens, ...)
◮ A simple group (Tits, Steinberg, Hertzig) ◮ Twisted compositions (F4, Springer) ◮ Trialitarian algebras (KMRT)
Trialitarian algebras are *algebras* classified by H1(F, PGO+
8 ⋊S3). They consist of
involution (A, σ) over L,
There is a *generic trialitarian algebra* whose center is a field
8 ⋊S3 (Parimala, Sridharan, K., 2000).
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 = PO+
8 ×
| S3. We
show how this can be translated to the invariants of the trialitarian Weyl group (S2)3 ×
| S4) × | S3 and then work out the consequences. As it turns
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
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
Fix a split torus T ⊂ PGO+
8 invariant under a trialitarian action
to get an induced trialitarian action on the Weyl group W = S3
2 ⋊ 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, ....
H1(F, S3
2 ⋊ 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 for for well situated involutions. Severi-Brauer varieties for well situated involutions (Tignol, K., 2011)