1. Let V be the category of finite-dimensional super vector spaces over R. Bysuper algebraI 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 aresuper Morita equivalentif they are equivalent objects in this bicategory. Equivalence classes [A] form an abelian monoid whose multiplication is given by the monoidal product. Thesuper Brauer groupof 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 = C_{1}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 i^{2}= -1. The opposite A* is similar except that there is an odd generator e satisfying e^{2}= 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 [C_{1}] is invertible. One manifestation of Bott periodicity is that [C_{1}] has order 8. We will soon see a very easy proof of this fact. A theorem of C.T.C. Wall is that [C_{1}] in fact generates the super Brauer group; I believe this can be shown by classifying super division algebras, as discussed below. 3. That [C_{1}] has order 8 is an easy calculation. Let C_{r}denote the r-fold tensor of C_{1}. C_{2}for instance has two super-commuting odd elements i, j satisfying i^{2}= j^{2}= -1; it follows that k := ij satisfies k^{2}= -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⟩. C_{3}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 C_{3}= H[e] mod (e^{2}- 1). Note this is the same as H tensor (C_{1})* where the H here is the quaternions viewed as a super algebra concentrated in degree 0 (i.e. ispurely bosonic). Then we see immediately that C_{4}= C_{3}tensor C_{1}is equivalent to purely bosonic H (since the C_{1}cancels (C_{1})* 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] = [C_{4}] has order 2! Hence [C_{1}] 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 A^{r}tensor (A*)^{s}, where A^{r}denotes r-fold tensor product of A = C_{1}. By the above calculation we see that Cliff(V, Q) is equivalent to C_{r-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: C_{0}~ R C_{1}~ R + Re, e^{2}= -1 C_{2}~ C + Ce, e^{2}= -1, ei = -ie C_{3}~ H + He, e^{2}= 1, ei = ie, ej = je, ek = ke C_{4}~ H C_{5}~ H + He, e^{2}= -1, ei = ie, ej = je, ek = ke C_{6}~ C + Ce, e^{2}= 1, ei = -ie C_{7}~ R + Re, e^{2}= 1 All the super algebras on the right are in factsuper division algebras, i.e. super algebras in which every nonzero homogeneous element is invertible. To prove Wall's result that [C_{1}] 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 e^{2}:= <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 bereal. 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 e^{2}- 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