Vector Cross Products

The classical result that r-fold vector cross products exist only for d-dimensional vector spaces with r = 1 and d even; r = 2 and d = 3 or 7; r = 3 and d = 8; and r = d − 1 for arbitrary d will be explained. Vector cross products will then be used to construct exceptional Lie superalgebras. 1. Bilinear vector cross products We all are familiar with the usual vector cross product × in R3, which satisfies:  u× v is bilinear, u× v ⊥ u, v, (so (u× v) · w is skew symmetric, and so is u× v) (u× v) · (u× v) = ∣∣ u·u u·v v·u v·v ∣∣ Definition 1. Let V be a d-dimensional vector space over a field F of characteristic 6= 2, endowed with a nondegenerate symmetric bilinear form (. | .). A bilinear map × : V × V → V is called a vector cross product if it satisfies the following conditions: (u× v | u) = (u× v) | v) = 0, (1) (u× v | u× v) = ∣∣∣∣(u | u) (u | v) (v | u) (v | v) ∣∣∣∣ , (2) for any u, v ∈ V . Our main purpose in this talk is to provide a proof of the following result: Theorem 1. Let × be a vector cross product on the vector space V . Then dimV = 1, 3 or 7. It is interesting to note that in 1943, Beno Eckmann gave a proof of this result for real euclidean spaces, but where the map × is not supposed to be bilinear, but continuous (which is manifestly a weaker condition). His proof used algebraic topology. It was in 1967 when R.B. Brown and A. Gray gave the first proof of the result above (actually, of an extension of it we will consider later on). A completely new and surprising proof was given in 1996 by M. Rost (later simplified by K. Meyberg in 2002). This proof is completely elementary, but it is valid only over fields of characteristic 0: