Sugawara construction. Commutator of a set of screenings Given a holomorphic field with values in , and a set of vectors , a W-algebra may be defined as the set of polynomials of and its derivatives that commute with the screening charges . If the vectors are the simple roots of a Lie algebra , the resulting W-algebra coincides with an algebra that is obtained from by Drinfeld-Sokolov reduction. The W(N) algebras For any integer , the W(N) algebra is a W-algebra which is generated by meromorphic fields of dimensions . The W(2) algebra coincides with the Virasoro algebra. Construction The W(N) algebra is obtained by Drinfeld-Sokolov reduction of the affine Lie algebra . The embeddings are parametrized by the integer partitions of , interpreted as decompositions of the fundamental representation of into representations of . The set of dimensions of the generators of the resulting W-algebra is such that where is the -dimensional irreducible representation of . The trivial partition corresponds to the W(N) algebra, while corresponds to itself. In the case , the partition leads to the Bershadsky-Polyakov algebra, whose generating fields have the dimensions . Properties The central charge of the W(N) algebra is given in terms of the level of the affine Lie algebra by in notations where the central charge of the affine Lie algebra is It is possible to choose a basis such that the commutation relations are invariant under . While the Virasoro algebra is a subalgebra of the universal enveloping algebra of , the W(N) algebra with is not a subalgebra of the universal enveloping algebra of . Example of the W(3) algebra The W(3) algebra is generated by the generators of the Virasoro algebra , plus another infinite family of generators . The commutation relations are where is the central charge, and we define The field is such that . Representation theory Highest weight representations A highest weight representation of a W-algebra is a representation that is generated by a primary state: a vector such that for some numbers called the charges, including the conformal dimension . Given a set of charges, the corresponding Verma module is the largest highest-weight representation that is generated by a primary state with these charges. A basis of the Verma module is where is the set of ordered tuples of strictly positive integers of the type with , and . Except for itself, the elements