W-algebra

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

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 consisting of a reductive Lie algebra 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 -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 .

Sources

  • de Boer, Jan; Tjin, Tjark (1993), "Quantization and representation theory of finite W algebras", Communications in Mathematical Physics, 158 (3): 485–516, arXiv:hep-th/9211109, Bibcode:1993CMaPh.158..485D, doi:10.1007/bf02096800, ISSN 0010-3616, MR 1255424, S2CID 204933347
  • Bouwknegt, P.; Schoutens, K., eds. (1995), W-symmetry, Advanced Series in Mathematical Physics, 22, River Edge, New Jersey: World Scientific Publishing Co., doi:10.1142/2354, ISBN 978-981021762-4, MR 1338864
  • Bouwknegt, Peter; Schoutens, Kareljan (1993), "W symmetry in conformal field theory", Physics Reports. A Review Section of Physics Letters, 223 (4): 183–276, arXiv:hep-th/9210010, Bibcode:1993PhR...223..183B, doi:10.1016/0370-1573(93)90111-P, ISSN 0370-1573, MR 1208246, S2CID 118959569
  • Brown, Jonathan, Finite W-algebras of Classical Type (PDF)
  • Dickey, L. A. (1997), "Lectures on classical W-algebras", Acta Applicandae Mathematicae, 47 (3): 243–321, doi:10.1023/A:1017903416906, ISSN 0167-8019, S2CID 118573600
  • Gan, Wee Liang; Ginzburg, Victor (2002), "Quantization of Slodowy slices", International Mathematics Research Notices, 2002 (5): 243–255, arXiv:math/0105225, doi:10.1155/S107379280210609X, ISSN 1073-7928, MR 1876934, S2CID 13895488
  • Losev, Ivan (2010), "Quantized symplectic actions and W-algebras", Journal of the American Mathematical Society, 23 (1): 35–59, arXiv:0707.3108, Bibcode:2010JAMS...23...35L, doi:10.1090/S0894-0347-09-00648-1, ISSN 0894-0347, MR 2552248, S2CID 16211165
  • Pope, C.N. (1991), Lectures on W algebras and W gravity, Lectures given at the Trieste Summer School in High-Energy Physics, August 1991, arXiv:hep-th/9112076, Bibcode:1991hep.th...12076P
  • Wang, Weiqiang (2011). "Nilpotent orbits and finite W-algebras". In Neher, Erhard; Savage, Alistair; Wang, Weiqiang (eds.). Geometric representation theory and extended affine Lie algebras. Fields Institute Communications Series. Volume 59. Providence, RI. pp. 71–105. arXiv:0912.0689. Bibcode:2009arXiv0912.0689W. ISBN 978-0-8218-5237-8. MR 2777648.
  • Watts, Gerard M. T. (1997), "W-algebras and their representations" (PDF), in Horváth, Zalán; Palla, László (eds.), 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 0829902
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.