AAC Mini Course

Abstract. The goal of this course is the introduction of the basic properties of the classical composition algebras (that is, those algebras which are anal-ogous to the real, complex, quaternion or octonion numbers), and how these basic properties are enough to get all the possible gradings on them. Then a new class of (non unital) composition algebras will be defined and studied, the so called symmetric composition algebras. Finally, the gradings on these two families of composition algebras will be shown to induce some interesting gradings on the exceptional simple Lie algebras. 1. Unital composition algebras. The Cayley-Dickson process. Composition algebras constitute a generalization of the classical algebras of the real R, complex C, quaternion H (1843), and octonion numbers O (1845). Definition 1.1. A composition algebra (over a field F) is a not necessarily associa-tive algebra C, endowed with a nondegenerate quadratic form (the norm) q: C → F (i.e., the bilinear form q(x, y) = q(x + y) − q(x) − q(y) is nondegenerate) which is multiplicative: q(xy) = q(x)q(y) ∀x, y ∈ C. The unital composition algebras will be called Hurwitz algebras Easy consequences: • q(xy, xz) = q(x)q(y, z) = q(yx, zx) ∀x, y, z. (lx and rx are similarities of norm q(x).) • q(xy, tz) + q(ty, xz) = q(x, t)q(y, z) ∀x, y, z, t. Assume now that C is unital: • t = 1 ⇒ q(xy, z) = q(y, (q(x, 1)1 − x)z) = q(y, x̄z) (x ̄ = q(x, 1)1 − x is an order 2 orthogonal map). That is: l∗x = lx̄, r x = rx̄. Then lxlx ̄ = rxrx ̄ = q(x)id, and applied to 1 this gives: x2 − q(x, 1)x+ q(x)1 = 0, ∀x (quadratic algebras