In addition to the two reasonably well-known categories SuperVectR and SuperVectC of real and complex super vector spaces, each of which is monoidally equivalent to corresponding category of Z/2-graded vector spaces but with the Koszul sign rules, there is a third much less well-known symmetric monoidal category that I like to call the quaternionic super vector spaces SuperVectH. It appears, among other places, when studying the statistics of a certain type of pinor.
As a category,
SuperVectH=VectR⊕ModH
The monoidal structure is a bit funny, using the Morita equivalence
H⊗H≃R. Here is a description of it. Recall that the usual Galois correspondence between
R and
C identifies
VectR with the category of complex vector spaces
VC equipped with an antilinear involution, i.e.
φ:VC→V∗C,
φ∗φ=1. Similarly, one can identify
ModH with the category of complex vector spaces
VC equipped with an antilinear "antiinvolution", i.e.
φ:VC→V∗C,
φ∗φ=−1. (By definition,
V∗=V as real vector spaces, but the
C-action is that
λ∈C acting on
v∈V∗ is given by the action
vλ∗ in
V. So
φ∗=φ as real linear maps, but I'm thinking of them as
C-linear in two different ways.) Using this, you can identify
SuperVectH with the category of complex supervector spaces equipped with an antilinear map that squares to
1 on the even part and to
−1 on the odd part. If you check carefully, you'll see that the tensor product (in
SuperVectC) of two such objects is naturally such an object, and in this way you can recover the symmetric monoidal structure on
SuperVectH.
In any sufficiently nice linear category (and SuperVectH is plenty nice for these purposes) you can develop a theory of associative algebras, bimodules, and Morita equivalence. Among other things, you can define a Brauer group for your given category whose elements are Morita equivalence classes of Morita-invertible algebras. (I.e. the group of units of the monoid whose objects are Morita equivalence classes of algebras and whose multiplication is tensor product.)
It is well known that the Brauer groups in this sense of SuperVectC and SuperVectR are respectively Z/2 and Z/8, and that this is closely related to periodicity in various K-theories.
Question: What is the Brauer group of SuperVectH? What are the simple representatives of the elements? What "K-theory" is it related to?
This post imported from StackExchange MathOverflow at 2016-02-04 18:36 (UTC), posted by SE-user Theo Johnson-Freyd