  Let $C$ be a simply laced generalized Cartan matrix. Given an element $b$ of
the generalized braid semigroup related to $C$, we construct a collection of
mutation-equivalent quivers with potentials. A quiver with potential in such a
collection corresponds to an expression of $b$ in terms of the standard
generators. For two expressions that differ by a braid relation, the
corresponding quivers with potentials are related by a mutation.
  The main application of this result is a construction of a family of $CY_3$
$A_\infty$-categories associated to elements of the braid semigroup related to
$C$. In particular, we construct a canonical up to equivalence $CY_3$
$A_\infty$-category associated to quotient of any Double Bruhat cell
$G^{u,v}/{\rm Ad} H$ in a simply laced reductive Lie group $G$.
  We describe the full set of parameters these categories depend on by defining
a 2-dimensional CW-complex and proving that the set of parameters is identified
with second cohomology group of this complex.
