W-algebra

For W*-algebra, see Von Neumann algebra.

In mathematics, a W-algebra is a structure in conformal field theory related to generalizations of the Virasoro algebra. They were introduced by 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

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

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

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

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

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

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

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

References

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Wang, Weiqiang (2011), "Nilpotent orbits and finite W-algebras", in Neher, Erhard; Savage, Alistair; Wang, Weiqiang, Geometric representation theory and extended affine Lie algebras, Fields Inst. Communications, 59, Providence, R.I.: American Mathematical Society, pp. 71–105, arXiv:0912.0689, ISBN 978-0-8218-5237-8, MR 2777648
Watts, Gerard M. T. (1997), "W-algebras and their representations", in Horváth, Zalán; Palla, László, Conformal field theories and integrable models (Budapest, 1996), Lecture Notes in Phys., 498, Berlin, New York: Springer-Verlag, pp. 55–84, doi:10.1007/BFb0105278, ISBN 978-3-540-63618-2, MR 1636798
Zamolodchikov, A. B. (1985), "Infinite extra symmetries in two-dimensional conformal quantum field theory", Akademiya Nauk SSSR. Teoreticheskaya i Matematicheskaya Fizika (in Russian), 65 (3): 347–359, ISSN 0564-6162, MR 829902
Brown, Jonathan, Finite W-algebras of Classical Type (PDF)

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