There is a homomorphism from the group of (isomorphism classes of) self-equivalences of a triangulated category to the automorphism group of its Grothendieck group. Is this homomorphism surjective? If not, is there a calculable obstruction? Are there some natural classes of triangulated categories for which it is surjective? In any case, what would be a good reference for this question?



Comparing self-equivalences of a triangulated category and automorphisms of its Grothendieck group
One obstruction is that if all Hom-sets are finite dimensional vector spaces, and for all objects $X$ and $Y$, $\text{Hom}(X,Y[i])=0$ for all but finitely many $i$, then any self-equivalence must preserve the bilinear form
$$\langle [X],[Y]\rangle =\sum\_i(-1)^i\dim\text{Hom}(X,Y[i])$$
on the Grothendieck group.


