Abdenacer Makhlouf - Algebra and Applications 1

Consider the spin group :

For any u ∈ Spin( , n),

for some such that

for any x , , where χ u( x ) = uxu −1gives the natural representation of Spin( , n), while give the two half-spin representations, and the formula above links the three of them.

The last condition is equivalent to:

for any x , y , , where

and this has cyclic symmetry:

THEOREM 2.7.– Let ( , ∗, n) be an eight-dimensional symmetric composition algebra. Then:

Spin .

Moreover, the set of related triples (the set on the right hand side) has cyclic symmetry.

The cyclic symmetry on the right-hand side induces an outer automorphism of order 3 ( trialitarian automorphism ) of Spin( , n). Its fixed subgroup is the group of automorphisms of the symmetric composition algebra ( , ∗, n), which is a simple algebraic group of type G 2in the para-Hurwitz case, and of type A 2in the Okubo case if char .

The group(-scheme) of automorphisms of an Okubo algebra over a field of characteristic 3 is not smooth (Chernousov et al . 2013).

At the Lie algebra level, assume , and consider the associated orthogonal Lie algebra

The triality Lie algebra of ( , ∗, n) is defined as the following Lie subalgebra of (with componentwise bracket):

Note that the condition d 0( x ∗ y ) = d 1( x ) ∗ y + x ∗ d 2( y ) for any x , is equivalent to the condition

for any x , y , . But n( x ∗ y , z ) = n( y ∗ z , x ) = n( z ∗ x , y ). Therefore, the linear map:

is an automorphism of the Lie algebra .

THEOREM 2.8.– Let ( , ∗, n) be an eight-dimensional symmetric composition algebra over a field of characteristic ≠ 2. Then:

– Principle of local triality: the projection map:

is an isomorphism of Lie algebras.

– For any x, , the triple

belongs to , and is spanned by these elements. Moreover, for any a , b , x , :

PROOF.– Let us first check that tx , y∈ :