As you can tell from the title of this post, I am trying to drag John Baez over to our blog.
Let be the ring of quaternions, i.e., with the standard relations. Let -mod be the category of left -modules.
This has an obvious tensor structure (including duals), inherited from the category of 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 be a quaternion. Anyone who works with quaternions knows that there are two notions of trace. The naive trace, , is the trace of multiplication by on any irreducible -module, using the obvious tensor structure. But there is a better notion, the reduced trace, which is equal to . Similarly, there is a naive norm, , and there is a reduced norm .
This all makes me think that there is a subtle tensor category structure on -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 cohomology; one of which is the issue above about traces in quaternion algebras.