Maximal orders in nonassociative quaternion algebras

Abstract

Let K be a field, and let E be a separable quadratic extension of K, with K-conjugation XH X. Let A be a four-dimensional algebra over K of the form E @ EJ where Jx = XJ for x in E. Let J2 = b. If b were in K, we would have one of the usual quaternion algebras. In this paper we take the same definition but with b in E outside K. Our A then is still a division algebra, though it is nonassociative, and we call it a nonassociative quaternion algebra. Products involving a factor from E still satisfy associativity, and indeed this fact can be used to characterize such algebras abstractly [7]. They have been familiar examples of nonassociative division algebras for over half a century [2]. Now, let R be a Dedekind domain with fraction field K. As in the associative case, we can define an (R-)order in A to be an R-submodule M of A containing 1, having rank 4, and closed under multiplication. No one previously seems to have realized that the maximal orders in these algebras might have interesting properties. The corresponding question in ordinary quaternion algebras has been extensively studied [4,6], but the theory here turns out to be quite different. In Section 3 we shall classify the isomorphism classes of maximal orders containing S, the integral closure of