From: "Mark Sapir" Subject: Re: nonassociative division algebras Date: 9 Sep 2000 15:00:13 -0500 Newsgroups: sci.math.research Summary: [missing] "John Baez" wrote in message news:8p1aok$o5e$1@Urvile.MSUS.EDU... > Let me define a "division algebra" to be a finite-dimensional real > vector space A with a bilinear map m: A x A -> A called "multiplication", > such that: 1) left or right multiplication by any nonzero element of A > is one-to-one (and thus onto), 2) there exists an element 1 in A that > is both the left and right unit for this multiplication. > > Let's say a division algebra "has multiplicative inverses" if > for every nonzero element a there is an element b with ab = ba = 1. > > Are there division algebras without multiplicative inverses? > > Of course, such a thing must be nonassociative, and it must not > be the octonions. > Take the algebra of quaternions and make, say, j*k=2i instead of i, that is the multiplication table is as in the quaternion algebra except for j*k. The resulting (non-associative) algebra will be a division algebra (easy to check, if you want I can explain how). The element a=i+j+k will have a right inverse b=-1/3 i - 5/12 j - 1/4 k+1/12, that is ab=1, but ba <> 1. So this right inverse is not a left inverse. Mark Sapir