\(W\)-algebras extending \(\widehat{\mathfrak{g}\mathfrak{l}}(1| 1)\). (English) Zbl 1287.81055
Dobrev, Vladimir (ed.), Lie theory and its applications in physics. IX international workshop. Based on the 9th workshop on Lie theory and its applications in physics, Varna, Bulgaria, June 20–26, 2011. Tokyo: Springer (ISBN 978-4-431-54269-8/hbk; 978-4-431-54270-4/ebook). Springer Proceedings in Mathematics & Statistics 36, 349-367 (2013).
These notes describe a family of some kind of extensions of the affine super Lie algebra \(\widehat{\mathfrak{gl}}(1|1)\), which can be realized algorithmically using the fusion rules of \({\mathfrak{gl}}(1|1)\) that give rise an infinite family of simple currents. Based on a previous work, the authors perform as an example these computations up to a certain order using a free field realization and study the \(W\)-algebras. In particular, they show that for certain families of parameters, these algebras contain a bosonic subalgebra and it is conjectured that this subalgebra equals the \(W^{(2)}_{N}\) algebra of Feigin and Semikhatov.
For the entire collection see [Zbl 1266.00027].
For the entire collection see [Zbl 1266.00027].
Reviewer: Gastón Andrés García (Córdoba)
MSC:
81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
17B81 | Applications of Lie (super)algebras to physics, etc. |
46L60 | Applications of selfadjoint operator algebras to physics |