  W-algebras are a class of non-commutative algebras related to the classical
universal enveloping algebras. They can be defined as a subquotient of U(g)
related to a choice of nilpotent element e and compatible nilpotent subalgebra
m. The definition is a quantum analogue of the classical construction of
Hamiltonian reduction.
  We define a quantum version of Hamiltonian reduction by stages and use it to
construct intermediate reductions between different W-algebras U(g,e) in type
A.This allows us to express the W-algebra U(g,e') as a subquotient of U(g,e)
for nilpotent elements e' covering e. It also produces a collection of
(U(g,e),U(g,e'))-bimodules analogous to the generalised Gel'fand-Graev modules
used in the classical definition of the W-algebra; these can be used to obtain
adjoint functors between the corresponding module categories.
  The category of modules over a W-algebra has a full subcategory defined in a
parallel fashion to that of the Bernstein-Gel'fand-Gel'fand (BGG) category O;
this version of category O(e) for W-algebras is equivalent to an infinitesimal
block of O by an argument of Mili\v{c}i\'{c} and Soergel. We therefore
construct analogues of the translation functors between the different blocks of
O, in this case being functors between the categories O(e) for different
W-algebras U(g,e). This follows an argument of Losev, and realises the category
O(e') as equivalent to a full subcategory of the category O(e) where e' is
greater than e in the refinement ordering.
