Discretized Virasoro algebra. (English) Zbl 1268.81088
Fuchs, Jürgen (ed.) et al., Noncommutative geometry and representation theory in mathematical physics. Satellite conference to the fourth European congress of mathematics, July 5–10, 2004, Karlstad, Sweden. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3718-4/pbk). Contemporary Mathematics 391, 59-67 (2005).
From the text: The Virasoro algebra (a central extension of the algebra of diffeomorphisms of the circle) got a prominent role in modern mathematical physics. Here I shall describe its deformation, which appears naturally after discretization of the circle. …
Section 1 is a reminder of the connection between the Virasoro algebra and the KdV hierarchy. Discretization and quantization, performed in Section 2, leads to the construction of a discrete deformation of the Virasoro algebra in Section 3. The highlight there is the appearance of Volkov’s group, associated with this deformation. Section 4 is devoted to the discussion of some functional-analytic aspects of discretization giving a simple explanation of the pertinent duality in terms of a modular double.
For the entire collection see [Zbl 1078.17001].
Section 1 is a reminder of the connection between the Virasoro algebra and the KdV hierarchy. Discretization and quantization, performed in Section 2, leads to the construction of a discrete deformation of the Virasoro algebra in Section 3. The highlight there is the appearance of Volkov’s group, associated with this deformation. Section 4 is devoted to the discussion of some functional-analytic aspects of discretization giving a simple explanation of the pertinent duality in terms of a modular double.
For the entire collection see [Zbl 1078.17001].
MSC:
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 17B68 | Virasoro and related algebras |
| 81T25 | Quantum field theory on lattices |
