“W-algebra”的意思、由来-开放百科全书

词条

W-algebra

释义

Classical W-algebras
Quantum W-algebras
Finite W-algebras
References

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 ({{harvtxt|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.

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

{{harvtxt|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

{{harvtxt|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 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 (e,h,f). The eigenspace decomposition of ad(h) induces a -grading on g:

Define a character (i.e. a homomorphism from g to the trivial 1-dimensional Lie algebra) 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 (g,e) and is usually denoted .

References

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3 : Conformal field theory|Integrable systems|Representation theory

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