Algebra and Applications 1. Abdenacer Makhlouf

      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 G₂ in the para-Hurwitz case, and of type A₂ in 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 d₀(x ∗ y) = d₁(x) ∗ y + x ∗ d₂(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} ∈ :

      and hence

      Also and (adjoint relative to the norm n), but RxLx = n(x)id, so RxLy + RyLx = n(x, y)id and hence , so that , and too. Therefore, .

      Since the Lie algebra is spanned by the σx_{, y} ’s, it is clear that the projection π₀ is surjective (and hence so are π₁ and π₂). It is not difficult to check that ker π₀ = 0, and therefore, π₀ is an isomorphism.

      Finally, the formula [ta, b, tx_{, y}] = tσa, b_{(x), y} + tx, σa, b_(y) follows from the “same” formula for the σ’s and the fact that π₀ is an isomorphism. □

      Given two symmetric composition algebras (, ∗, n) and , consider the vector space:

      where is just a copy of (i = 0, 1, 2) and we write , instead of and for short. Define now an anticommutative bracket on by means of:

Скачать книгу