As you can tell from the title of this post, I am trying to drag John Baez over to our blog.
Let ![Q]()
be the ring of quaternions, i.e., ![\mathbb{R} \langle i,j,k \rangle]()
with the standard relations. Let ![Q]()
-mod be the category of left ![Q]()
-modules. This has an obvious tensor structure (including duals), inherited from the category of ![\mathbb{R}]()
vector spaces. Actually, that structure doesn’t quite work; I’ll leave to you good folks to work out what I should have said.
Let ![q=a+bi+cj+dk]()
be a quaternion. Anyone who works with quaternions knows that there are two notions of trace. The naive trace, ![4a]()
, is the trace of multiplication by ![a]()
on any irreducible ![Q]()
-module, using the obvious tensor structure. But there is a better notion, the reduced trace, which is equal to ![2a]()
. Similarly, there is a naive norm, ![(a^2+b^2+c^2+d^2)^2]()
, and there is a reduced norm ![a^2+b^2+c^2+d^2]()
.
This all makes me think that there is a subtle tensor category structure on ![Q]()
-mod, other than the obvious one, for which these are the trace and norm in the categorical sense. Can someone spell out the details for me?
By the way, a note about why I am asking. I am reading Milne’s excellent notes on motives, and I therefore want to understand the notion of a non-neutral Tannakian category (page 10). As I understand it, this notion allows us to evade some of the standard problems in defining characteristic ![p]()
cohomology; one of which is the issue above about traces in quaternion algebras.
