Algebra and Applications 1. Abdenacer Makhlouf

Algebra and Applications 1

characteristic 3, which completes the classification of symmetric composition algebras over fields, is as follows (Elduque (1997), see also Chernousov et al. (2013)):

THEOREM 2.6.– Any symmetric composition algebra (, ∗, n) over a field of characteristic 3 is either:

– a para-Hurwitz algebra. Two such algebras are isomorphic if and only if the associated Hurwitz algebras are too;

– a two-dimensional algebra with a basis {u, v} and multiplication given by

for a non-zero scalar . These algebras do not contain idempotents and are twisted forms of the para-Hurwitz algebras.

Algebras corresponding to the scalars λ and λʹ are isomorphic if and only if .

– Isomorphic to for some 0 ≠ α, . Moreover, is isomorphic or anti-isomorphic to if and only if .

A more precise statement for the isomorphism condition in the last item is given in (Elduque 1997). A key point in the proof of this theorem is the study of idempotents on Okubo algebras. If there are non-zero idempotents, then these algebras are Petersson algebras. The most difficult case appears in the absence of idempotents. This is only possible if the ground field is not perfect.

2.5. Triality

The importance of symmetric composition algebras lies in their connections with the phenomenon of triality in dimension 8, related to the fact that the Dynkin diagram D₄ is the most symmetric one. The details of much of what follows can be found in (Knus et al. (1998), Chapter VIII).

Let (, ∗, n) be an eight-dimensional symmetric composition algebra over a field , that is, is either a para-Hurwitz algebra or an Okubo algebra. Write Lx(y) = x ∗ y = Ry(x) as usual. Then, due to theorem 2.3, Lx Rx = RxLx = n(x)id for any so that, inside , we have

Therefore, the map

extends to an isomorphism of associative algebras with involution:

where (, n) is the Clifford algebra on the quadratic space (, n), τ is its canonical involution (τ (x) = x for any ) and σn_⊥n is the orthogonal involution on induced by the quadratic form where the two copies of S are orthogonal and the restriction on each copy coincides with the norm. The multiplication in the Clifford algebra will be denoted by juxtaposition.

Consider the spin group:

For any u ∈ Spin(, n),

for some such that

for any x, , where χ_u(x) = uxu⁻¹ gives 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 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

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