A W-algebra is an algebraic structure constructed from a universal enveloping algebra and a nilpotent element of the underlying Lie algebra; more precisely it can be constructed by reducing the universal enveloping algebra in a manner analogous to Hamiltonian reduction of Poisson varieties, known as quantum Hamiltonian reduction. In fact, W-algebras form a quantisation of the ring of functions on the appropriate Slodowy slices corresponding to the chosen nilpotent elements. More generally, we will show that W-algebras corresponding to more regular nilpotent elements can be obtained from more singular W-algebras using quantum Hamiltonian reduction, and mention some applications this has to categorification of tensor products of simple representations of
sl2.