W-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

• 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
• Bouwknegt, P.; Schoutens, K., eds. (1995), W-symmetry, Advanced Series in Mathematical Physics 22, River Edge, New Jersey: World Scientific Publishing Co. Inc., ISBN 9789810217624, MR 1338864
• 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
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• 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
• 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
• 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)