An octonionic formulation of the M-theory algebra

A bstract We give an octonionic formulation of the N = 1 supersymmetry algebra in D = 11, including all brane charges. We write this interms of a novel outer product, which takes a pair of elements of the division algebra A and returns a real linear operator on A . More generally, with this product comes the power to rewrite any linear operation on ℝ n ( n =1 , 2 , 4 , 8) in terms of multiplication in the n -dimensional division algebra A . Finally, we consider the reinterpretation of the D =11 supersymmetry algebra as an octonionic algebra in D =4 and the truncation to division subalgebras.