W-algebra


In conformal field theory and representation theory, a W-algebra is an 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

defines a W-algebra to be a meromorphic conformal field theory together with a distinguished set of generators satisfying various properties.
They can be constructed from a Lie 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

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 consisting of a reductive Lie algebra over the complex numbers and a nilpotent element e.
By the Jacobson-Morozov theorem, e is part of an sl2 triple. The eigenspace decomposition of ad induces a -grading on g:
Define a character by the rule, where 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, we may define the following nilpotent subalgebra which acts on the universal enveloping algebra by the adjoint action.
The left ideal of the universal enveloping algebra generated by is invariant under this action. It follows from a short calculation that the invariants in under ad inherit the associative algebra structure from. The invariant subspace is called the finite W-algebra constructed from and is usually denoted.