Over a noncommutative algebra $A$ we have no problem in defining invertible bimodules (as in the book by Bass on algebraic $K$-theory) - corresponding to line bundles over topological spaces $X$ if $A=C(X)$. Further we can make these into a group (the Picard group) by taking the group operation $\otimes\_A$. However after quite a bit of searching I cannot find examples where the resulting Picard group is itself nonabelian. 


I would be very pleased if people could point out examples where Picard group is nonabelian - I assume that they do exist, since there is no obvious reason why it should be abelian (I think... I may be wrong...). Examples where $A$ is a star algebra would be especially welcome. 


This question came from considering the properties of `line bundles' over non commutative spaces, and noncommutative algebraic geometry. Apologies if it is well known in some circles!



Does a nonabelian Picard group exist?
By the Eilenberg-Watts theorem, the 2-group of invertible $(A, A)$-bimodules is naturally equivalent to the 2-group of automorphisms of $\text{Mod}(A)$. There is a natural map


$$\text{Aut}(A) \to \pi\_0 \text{Aut}(\text{Mod}(A))$$


whose kernel is the subgroup of inner automorphisms. So it suffices to find $A$ whose outer automorphism group is nonabelian, and these are plentiful. (This conclusion does not require the Eilenberg-Watts theorem to reach, but it's convenient to phrase things this way.)


For a commutative example, we can let $A$ be a Galois extension of some field with nonabelian Galois group. For a noncommutative example we can let $A$ be a matrix algebra over such a thing. 


