Triality: from Geometry to Cohomology M-A. Knus, ETH Zrich - PowerPoint PPT Presentation

Triality: from Geometry to Cohomology M-A. Knus, ETH Zürich Ramification in Algebra and Geometry at Emory Atlanta, May 18, 2011

Outline ◮ Introduction and some historical comments ◮ A different approach at triality ◮ David’s work on triality

I. Introduction and some history 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 ...”

Basic object : The algebraic group PGO + 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, ◮ PO 8 the group of similitudes of q , PO 8 = { f ∈ GL 8 | q � � f ( x ) = µ ( f ) q ( x ) } , where µ ( f ) ∈ F × is the multiplier of f . Any φ ∈ PO 8 acts on C 0 ( q ) , C 0 ( φ )( xy ) = µ − 1 φ ( x ) φ ( y ) and PGO + 8 = { φ ∈ PO 8 | C 0 ( φ ) | Z = Id Z } , Z = center of C 0 ( q ) .

Projective Geometry PGO 8 is the group of the geometry on the quadric Q 6 in P 7 defined by q = 0. There are two types of projective subspaces of Q 6 of maximal dimension 3: *solids of type I and solids of type II*. The group PGO + 8 is the subgroup of PGO 8 which respect the two types. Projective subspaces of Q 2 of maximal dimension 1 in P 3

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 Q 6 is isomorphic to a quadric Q 6 . II. Any proposition in the geometry of Q 6 [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 + 8 → Aut ( PGO + 8 ) → S 3 → 1 where the homomorphism PGO + 8 → Aut ( PGO + 8 ) is given by inner automorphisms 8 �→ Int ( g )( x ) = gxg − 1 ∈ Aut ( PGO + g ∈ PGO + 8 ) . and S 3 occurs as the automorphism group of the Dynkin diagram of D 4 : ❞ α 3 ✔ ✔ ❞ ❞ ❚ ❚ α 1 α 2 ❞ α 4

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 . τ : A �→ C induce σ, τ ∈ Aut PGO + ( n ) such � � σ : A �→ B , that σ 3 = 1, τ 2 = 1, � σ, τ � = S 3 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 K a = { x ∈ O | ax = 0 } and R a = { x ∈ O | xa = 0 } . I . II . II Geometric triality can be described as a �→ K a �→ R a �→ 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 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 D 4 with trialitarian action. G of type 1 , 2 D 4 , ⇒ G = PGO + ( n ) , I n norm of some octonion algebra. II There is a split exact sequence 1 → PGO + ( n ) → Aut PGO + ( n ) → S 3 → 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 and x ⋆ ( y ⋆ x ) = ( x ⋆ y ) ⋆ x = n ( x ) y b n ( x ⋆ y , z ) = b n ( 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 f 1 , f 2 ∈ GO + ( n ) such that µ ( f ) − 1 f ( x ⋆ y ) = f 2 ( x ) ⋆ f 1 ( y ) µ ( f 1 ) − 1 f 1 ( x ⋆ y ) = f ( x ) ⋆ f 2 ( y ) µ ( f 2 ) − 1 f 2 ( x ⋆ y ) = f 1 ( x ) ⋆ f ( y ) . One formula implies the two others. The pair ( f 1 , f 2 ) 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 ) = M 2 End F S . � � Corollary The map ρ ⋆ : PGO + ( n ) → PGO + ( n ) defined by f ∈ GO + ( n ) ρ ⋆ ([ f ]) = [ f 2 ] , is a trialitarian automorphism of PGO + ( n ) . Observe that ρ ⋆ satisfies ρ 2 ⋆ ([ f ]) = [ f 1 ] .

Symmetric compositions and projective geometry Let ( S , ⋆, n ) be a symmetric composition and let Q 6 = { 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 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 ρ 2 ⋆ ) by an inner automorphism, let ρ = Int � � [ f ] ◦ ρ ⋆ x ⋄ y = f f − 1 f f 1 f − 1 ( x ) ⋆ f − 1 ( y ) then ρ = ρ ⋄ with . � � 1 [Recall µ ( f ) − 1 f ( x ⋆ y ) = f 2 ( x ) ⋆ f 1 ( y ) ]. Remark ρ 3 = 1 implies f f 1 f 2 = Id S .

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 G 2 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 G 2 . Conclusion : For any such n there is at least one symmetric composition !

Symmetric compositions of type A 2 Let ω ∈ F , ω 3 = 1. I A central simple of degree 3 over F , A 0 = { x ∈ S | Trd ( x ) = 0 } . − 1 n ( x ) = − 1 x ⋆ y = xy − ω yx 6 Trd ( x 2 ) . 3 tr ( xy ) , 1 − ω ( A 0 , ⋆, n ) is a symmetric composition whose norm is of maximal index. II Aut F ( A 0 , ⋆ ) = PGL 1 ( 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 , B 0 = { x ∈ B | τ ( x ) = x , Trd ( x ) = 0 } , ⋆ and n as above. Then ( B 0 , ⋆, n ) is a symmetric composition whose norm is is a 3-Pfister form � � 3 , α, β � � , for some α, β ∈ F × . II Aut F ( A 0 , ⋆ ) = PGU 1 ( B , τ ) .