Suppose we have a semiorthogonal decomposition $\mathcal{D} = \langle \mathcal{A}, \mathcal{B} \rangle$, and suppose we know fully the autoequivalence groups $\mathrm{Aut}(\mathcal{A})$ and $\mathrm{Aut}(\mathcal{B})$. Then do we know $\mathrm{Aut}(\mathcal{D})$?



Autoequivalence group from semiorthogonal decomposition
No. For instance, if $D\_n$ is the derived category of representations of a quiver with two vertices and $n$ arrows then $D$ has a semiorthogonal decomposition with two components equivalent to derived categories of the base field (so this does not depend on $n$) , but the connected component of the group of autoequivalences of $D\_n$ is $\mathrm{PGL}\_n$.


