IV - Composition algebras

Summary

This chapter provides an in-depth study of composition algebras over commutative rings, which we carry out in the more general framework of conic algebras (called quadratic algebras or algebras of degree 2 by other authors). We present the Cayley–Dickson construction and define composition algebras as unital nonassociative algebras that are projective as modules and allow a non-singular quadratic form permitting composition. We use this construction to obtain first examples of octonion algebras more general than the Graves–Cayley octonions and to derive structure theorems for arbitrary composition algebras. Specializing, it is shown that all composition algebras of rank at least 2 over an LG ring arise from an appropriate quadratic étale algebra by the Cayley–Dickson construction. Other examples of octonion algebras are obtained using ternary hermitian spaces. We address the norm equivalence problem, which asks whether composition algebras are classified by their norms and has an affirmative answer over LG rings but not in general. After a short excursion into affine (group) schemes, we conclude the chapter by showing that arbitrary composition algebras are split by étale covers.