The Super Brauer Group and Super Division Algebras
Todd Trimble
April 27, 2005
1. Let V be the category of finite-dimensional super vector
spaces over R. By super algebra I mean a monoid in this
category. There's a bicategory whose objects are super
algebras A, whose 1-cells M: A --> B are left A- right B-
modules in V, and whose 2-cells are homomorphisms
between modules. This is a symmetric monoidal bicategory
under the usual tensor product on V.
A and B are super Morita equivalent if they are equivalent
objects in this bicategory. Equivalence classes [A] form
an abelian monoid whose multiplication is given by the
monoidal product. The super Brauer group of R is the
subgroup of invertible elements of this monoid.
If [B] is inverse to [A] in this monoid, then in particular
A tensor (-) can be considered left biadjoint to B tensor (-).
On the other hand, in the bicategory above we always have
a biadjunction
A tensor C --> D
-------------------
C --> A* tensor D
(essentially because left A-modules are the same as right
A*-modules, where A* denotes the super algebra opposite
to A). Since right biadjoints are unique up to equivalence,
we see that if an inverse to [A] exists, it must be [A*].
This can be sharpened: an inverse to [A] exists iff the
unit and counit
1 --> A* tensor A A tensor A* --> 1
are equivalences in the bicategory. Actually, one is an
equivalence iff the other is, because both of these
canonical 1-cells are given by the same A-bimodule,
namely the one given by A acting on both sides of
the underlying superspace of A (call it S) by multiplication.
Either is an equivalence if the bimodule structure map
A* tensor A --> Hom(S, S),
which is a map of super algebras, is an isomorphism.
2. As an example, let A = C1 be the Clifford algebra
generated by the 1-dimensional space R with the usual
quadratic form Q(x) = |x|2, and Z/2-graded in the usual
way. Thus, the homogeneous parts of A are 1-dimensional
and there is an odd generator i satisfying i2 = -1. The
opposite A* is similar except that there is an odd generator
e satisfying e2 = 1. Under the map
A* tensor A --> Hom(S, S),
where we write S as a sum of even and odd parts R + Ri,
this map has a matrix representation
-1 0
e tensor i |--> 0 1
0 -1
1 tensor i |--> 1 0
0 1
e tensor 1 |--> 1 0
which makes it clear that this map is surjective and thus
an isomorphism. Hence [C1] is invertible.
One manifestation of Bott periodicity is that [C1]
has order 8. We will soon see a very easy proof of this
fact. A theorem of C.T.C. Wall is that [C1] in fact
generates the super Brauer group; I believe this can
be shown by classifying super division algebras, as
discussed below.
3. That [C1] has order 8 is an easy calculation. Let
Cr denote the r-fold tensor of C1. C2 for instance
has two super-commuting odd elements i, j satisfying
i2 = j2 = -1; it follows that k := ij satisfies
k2 = -1, and we get the usual quaternions, graded so that
the even part is the span ⟨1, k⟩ and the odd part is ⟨i, j⟩.
C3 has three super-commuting odd elements i, j, l, all
of which are square roots of -1. It follows that
e = ijl is an odd central involution (here "central"
is taken in the ungraded sense), and also that
i' = jl, j' = li, k' = ij satisfy the Hamiltonian equations
(i')2 = (j')2 = (k')2 = i'j'k' = -1,
so we have C3 = H[e] mod (e2 - 1). Note this is the
same as
H tensor (C1)*
where the H here is the quaternions viewed as a super
algebra concentrated in degree 0 (i.e. is purely bosonic).
Then we see immediately that C4 = C3 tensor C1 is
equivalent to purely bosonic H (since the C1 cancels
(C1)* in the super Brauer group).
At this point we are done: we know that conjugation on
(purely bosonic) H gives an isomorphism
H* --> H
hence [H]-1 = [H*] = [H], i.e. [H] = [C4] has order 2!
Hence [C1] has order 8.
4. All this generalizes to Clifford algebras: if a real quadratic
vector space (V, Q) has signature (r, s), then the super
algebra Cliff(V, Q) is isomorphic to Ar tensor (A*)s,
where Ar denotes r-fold tensor product of A = C1.
By the above calculation we see that Cliff(V, Q) is
equivalent to Cr-s where r-s is taken modulo 8.
For the record, then, here are the hours of the super
Clifford clock, where e denotes an odd element, and ~
denotes equivalence:
C0 ~ R
C1 ~ R + Re, e2 = -1
C2 ~ C + Ce, e2 = -1, ei = -ie
C3 ~ H + He, e2 = 1, ei = ie, ej = je, ek = ke
C4 ~ H
C5 ~ H + He, e2 = -1, ei = ie, ej = je, ek = ke
C6 ~ C + Ce, e2 = 1, ei = -ie
C7 ~ R + Re, e2 = 1
All the super algebras on the right are in fact super
division algebras, i.e. super algebras in which every
nonzero homogeneous element is invertible.
To prove Wall's result that [C1] generates the super
Brauer group, we need a lemma: any element in the super
Brauer group is the class of a super division algebra.
For this, we need the classification of super division algebras.
I'll take as known that the only associative division algebras
over R are R, C, H -- the even part A of an associative super
division algebra must be one of these cases. We can
express the associativity of a super algebra (with even part
A) by saying that the odd part M is an A-bimodule equipped
with a A-bimodule map pairing
< -, - >: M tensor_A M --> A
such that:
(**) a<b, c> = <a, b>c for all a, b, c in M.
If the super algebra is a super division algebra which is not
wholly concentrated in even degree, then multiplication by a
nonzero odd element induces an isomorphism
A --> M
and so M is 1-dimensional over A; choose a basis element e
for M.
The key observation is that for any a in A, there exists a
unique a' in A such that
ae = ea'
and that the A-bimodule structure forces (ab)' = a'b'. Hence we
have an automorphism (fixing the real field)
(--)': A --> A
and we can easily enumerate (up to isomorphism) the possibilities
for associative super division algebras over R:
(1) A = R. Here we can adjust e so that either e2 := <e, e> is
-1 or 1. The corresponding super division algebras occur at 1 o'clock
and 7 o'clock on the super Brauer clock.
(2) A = C. There are two R-automorphisms C --> C. In the case
where the automorphism is conjugation, condition (**) for super
associativity gives <e, e>e = e<e, e> so that <e, e> must be
real. Again e can be adjusted so that <e, e> = -1 or 1. These
possibilities occur at 2 o'clock and 6 o'clock on the super Brauer
clock.
For the identity automorphism, we can adjust e so that <e, e>
is 1. This gives the super algebra C[e] mod e2 - 1 (where e
commutes with elements in C). This does not occur on the
super Brauer clock over R. However, it does generate the super
Brauer group over C (which is of order two).
(3) A = H. Here R-automorphisms H --> H are given by
h |--> xhx-1 for x in H. In other words
he = exhx-1
whence ex commutes with all elements of H (i.e. we can
assume wlog that the automorphism is the identity). The
properties of the pairing guarantee that h<e, e> = <e, e>h
for all h in H, so <e, e> is real and again we can adjust
e so that <e, e> = 1 or -1. These cases occur at 3 o'clock
and 5 o'clock on the super Brauer clock.
This appears to be a complete (even if a bit pedestrian)
analysis.
© 2005 Todd H. Trimble