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 ${\displaystyle ({\mathfrak {g}},e)}$ 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

• 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