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 (Zamolodchikov (1985)), 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.

For W*-algebra, see Von Neumann algebra.

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 & Schoutens (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 ${\displaystyle ({\mathfrak {g}},e)}$ consisting of a reductive Lie algebra ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle \mathbb {Z} }$-grading on g:

${\displaystyle {\mathfrak {g}}=\bigoplus {\mathfrak {g}}(i).}$

Define a character ${\displaystyle \chi }$ (i.e. a homomorphism from g to the trivial 1-dimensional Lie algebra) by the rule ${\displaystyle \chi (x)=\kappa (e,x)}$, where ${\displaystyle \kappa }$ denotes the Killing form. This induces a non-degenerate anti-symmetric bilinear form on the -1 graded piece by the rule:

${\displaystyle \omega _{\chi }(x,y)=\chi ([x,y]).}$

After choosing any Lagrangian subspace ${\displaystyle l}$, we may define the following nilpotent subalgebra which acts on the universal enveloping algebra by the adjoint action.

${\displaystyle {\mathfrak {m}}=l+\bigoplus _{i\leq -2}{\mathfrak {g}}(i).}$

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

Sources

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