W algebra

Classical W-algebras

Performing classical Drinfeld-Sokolov reduction on a Lie algebra provides the Poisson bracket on this algebra.

Quantum W-algebras

Bouwknegt (1993) defines 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 (2011) 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 ( g , e ) consisting of a reductive Lie algebra g over the complex numbers and a nilpotent element e. By the Jacobson-Morozov theorem, e is part of an sl₂ triple (e,h,f). The eigenspace decomposition of ad(h) induces a Z -grading on g:

Define a character χ (i.e. a homomorphism from g to the trivial 1-dimensional Lie algebra) by the rule χ ( x ) = κ ( e , x ) , where κ 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 l , we may define the following nilpotent subalgebra which acts on the universal enveloping algebra by the adjoint action.

The left ideal I of the universal enveloping algebra U ( g ) generated by { x − χ ( x ) : x ∈ m } is invariant under this action. It follows from a short calculation that the invariants in U ( g ) / I under ad ( m ) inherit the associative algebra structure from U ( g ) . The invariant subspace ( U ( g ) / I ) ad ( m ) is called the finite W-algebra constructed from (g,e) and is usually denoted U ( g , e ) .