Algebra and Applications 1 - Abdenacer Makhlouf

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alt="image"/>, ∗, n) is defined as the following Lie subalgebra of image (with componentwise bracket):

image

      Note that the condition d0(x ∗ y) = d1(x) ∗ y + x ∗ d2(y) for any x, image is equivalent to the condition

image

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

image

      is an automorphism of the Lie algebra image.

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

       – Principle of local triality: the projection map:

 image

      is an isomorphism of Lie algebras.

       – For any x, , the triple

 image

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

image

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

image

      and hence

image

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

      Since the Lie algebra image 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 (image, ∗, n) and image, consider the vector space:

image

      where image is just a copy of image (i = 0, 1, 2) and we write image, image instead of image and image for short. Define now an anticommutative bracket on image 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 image. With the bracket above, image is a Lie algebra and, if image and image denote symmetric composition algebras of dimension r and s, then the Lie algebra image is a (semi) simple Lie algebra whose type is given by Freudenthal’s magic square:

image

      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)).

      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

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