W-algebra

In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples.
There are at least three different but related notions called W-algebras: classical W-algebras, quantum W-algebras, and finite W-algebras.
Classical W-algebras.
Performing classical Drinfeld-Sokolov reduction on a Lie algebra provides the Poisson bracket on this algebra.
Quantum W-algebras.
Bouwknegt and Schoutens define a (quantum) W-algebra to be a meromorphic conformal field theory (roughly a vertex operator algebra) together with a distinguished set of generators satisfying various properties.
They can be constructed from a Lie (super)algebra by quantum Drinfeld–Sokolov reduction. Another approach is to look for other conserved currents besides the Stress–energy tensor in a similar manner to how the Virasoro algebra can be read off from the expansion of the stress tensor.
Finite W-algebras.
Wang compares several different definitions of finite W-algebras, which are certain associative algebras associated to nilpotent elements of semisimple Lie algebras.
The original definition, provided by Alexander Premet, starts with a pair formula_1 consisting of a reductive Lie algebra formula_2 over the complex numbers and a nilpotent element e.
By the Jacobson-Morozov theorem, e is part of a sl2 triple (e, h, f). The eigenspace decomposition of ad(h) induces a formula_3-grading on g:
Define a character formula_5 (i.e. a homomorphism from g to the trivial 1-dimensional Lie algebra) by the rule formula_6, where formula_7 denotes the Killing form. This induces a non-degenerate anti-symmetric bilinear form on the -1 graded piece by the rule:
After choosing any Lagrangian subspace formula_9, we may define the following nilpotent subalgebra which acts on the universal enveloping algebra by the adjoint action.
The left ideal formula_11 of the universal enveloping algebra formula_12 generated by formula_13 is invariant under this action. It follows from a short calculation that the invariants in formula_14 under adformula_15 inherit the associative algebra structure from formula_12. The invariant subspace formula_17 is called the finite W-algebra constructed from (g, e) and is usually denoted formula_18.