From: "Mark Sapir" <msapir@home.com>
Subject: Re: nonassociative division algebras
Date: 9 Sep 2000 15:00:13 -0500
Newsgroups: sci.math.research
Summary: [missing]

"John Baez" <baez@newmath.UCR.EDU> 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