  In this article we construct three explicit natural subgroups of the
Brauer-Picard group of the category of representations of a finite-dimensional
Hopf algebra. In examples the Brauer Picard group decomposes into an ordered
product of these subgroups, somewhat similar to a Bruhat decomposition.
  Our construction returns for any Hopf algebra three types of braided
autoequivalences and correspondingly three families of invertible bimodule
categories. This gives examples of so-called (2-)Morita equivalences and
defects in topological field theories. We have a closer look at the case of
quantum groups and Nichols algebras and give interesting applications. Finally,
we briefly discuss the three families of group-theoretic extensions.
