Algebra and Applications 1. Abdenacer Makhlouf

, ∗, 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:

       – the Lie bracket in , which thus becomes a Lie subalgebra of ;

       – [(d0, d1, d2), ιi(x ⊗ x′)] = ιi(di(x) ⊗ x′);

       –

       – [ιi(x ⊗ x′), ιi+1(y ⊗ y′)] = ιi+2((x ∗ y) ⊗ (x′ ★ y′)) (indices modulo 3);

       –

      THEOREM 2.9 (Elduque (2004)).– Assume char . With the bracket above, is a Lie algebra and, if and denote symmetric composition algebras of dimension r and s, then the Lie algebra is a (semi) simple Lie algebra whose type is given by Freudenthal’s magic square:

      Different versions of this result using Hurwitz algebras instead of symmetric composition algebras have appeared over the years (see Elduque (2004) and the references therein). The advantage of using symmetric composition algebras is that new constructions of the exceptional simple Lie algebras are obtained, and these constructions highlight interesting symmetries due to the different triality automorphisms.

      A few changes are needed for characteristic 3. Also, quite interestingly, over fields of characteristic 3 there are non-trivial symmetric composition superalgebras, and these can be plugged into the previous construction to obtain an extended Freudenthal’s magic square that includes some new simple finite dimensional Lie superalgebras (see Cunha and Elduque (2007)).

2.6. Concluding remarks

      It is impossible to give a thorough account of composition algebras in a few pages, so many things have had to be left out: Pfister forms and the problem of composition of quadratic