From: Dave Rusin Subject: Re: nonassociative division algebras Date: Wed, 6 Sep 2000 00:32:52 -0500 (CDT) To: baez@newmath.UCR.EDU Summary: [missing] I didn't post your message because I thought the answer was simple enough to respond on my own. Well, the answer _was_ simple, as I thought, but not for the reasons I thought -- it wasn't as simple as I expected to show that the answer was simple (huh?). Anyway, here's a response to: >Are there division algebras without multiplicative inverses? Sure. Well, "division algebra" means left- and right- multiplicative inverses exist; the question is whether they need to be equal, right? Division algebras are plentiful. In the right dimensions, they're hard to miss! Just look at 4-dimensional algebras over R. Pick a basis for such an algebra, say {e0, e1, e2, e3}. Then left-multiplication by each e_i determines a 4x4 matrix X_i, and these X_i completely describe the algebra. The four matrices define a division algebra iff det( \Sum x_i X_i ), a homogeneous quartic polynomial in four variables, has no zeros except at the origin, in other words, its minimum over S^3 is positive. That minimum depends continuously on the coefficients of the X_i, and so "minimum is positive" is an open condition. In particular, any small pertubation of the 64 matrix entries defining the quaternion algebra will give another division algebra. Pick something at random, and observe that the left- and right- inverses of e4 (say) are different. Putting this into practice, I got a toy example with Maple. Let A be the span of four elements 1, i, j, k where all 16 products of two of these are as given in the quaternion algebra EXCEPT that k*k= -1 + t i , where the real number t will be determined in a moment. Then multiplication by x0 1 + x1 i + x2 j + x3 k is represented by a 4x4 matrix whose determinant I work out to be 2 2 2 2 2 2 2 2 2 (x0 + x1 + x2 + x3 ) + x2 x3 (x0 + x1 + x2 + x3 ) t Note that on the unit sphere this is 1 + x2 x3 t, so as long as |t| < 1, the determinant will stay positive on the unit sphere, and hence (by scaling) positive everywhere except at 0+0i+0j+0k. So for such t, I have defined a division algebra. Now we check for the inverses of k: I get k * ( -t j - k ) = 1 ( +t j - k ) * k = 1 so the left- and right- inverses are different. dave