Generalized Drinfel'd-Sokolov hierarchies

Burroughs, Nigel John; Groot, Mark F. de; Hollowood, Timothy J.; Miramontes, J. Luis

doi:10.1007/bf02099045

Cited by 76 publications

(227 citation statements)

References 15 publications

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“…In conclusion, both the canonical and integrability formalisms agree and lead to the W (2) 3 symmetry algebra at any finite λ. The preservation of the W (2) 3 integrability structure under deformation was also noticed in [75,76]. stimulating discussions and useful comments.…”

Section: Bi-hamiltonian Structurementioning

confidence: 60%

$ \mathcal{W} $ symmetry and integrability of higher spin black holes

Compère

Song²

2013

J. High Energ. Phys.

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We obtain the asymptotic symmetry algebra of sl(3, R) × sl(3, R) Chern-Simons theory with Dirichlet boundary conditions for fixed chemical potential. These boundary conditions are obeyed by higher spin black holes. For each embedding of sl(2, R) into sl(3, R), we show that the asymptotic symmetry group is independent of the chemical potential. On the one hand, starting from AdS 3 in the principal embedding, we show that the W 3 × W 3 symmetry is preserved upon turning on perturbatively spin 3 chemical potentials. On the other hand, starting from AdS 3 in the diagonal embedding, we show that the W (2) 3 ×W (2) 3 symmetry is preserved upon turning on finite spin 3/2 chemical potentials. We also make connections between the canonical Lagrangian formalism and integrability methods based on the n = 3 KdV (Boussinesq) hierarchy.

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Section: Bi-hamiltonian Structurementioning

confidence: 60%

$ \mathcal{W} $ symmetry and integrability of higher spin black holes

Compère

Song²

2013

J. High Energ. Phys.

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“…Namely, we will derive the sl (2) 3 hierarchy, the one associated with the conformal algebra W (2) 3 [4,1,20], along the lines of [5], showing that it coincides with the KP (2) 3 hierarchy previously introduced. We consider the Lie algebra sl 3 and its loop algebra G = L(sl 3 ), i.e., the space of C ∞ -maps from S 1 to sl 3 .…”

Section: The Hierarchies Kp (M)mentioning

confidence: 96%

“…It is not difficult to check that our parameters are a family equivalent to those already appeared in the literature [1,4] and have the right scaling dimensions. The reduced brackets on N can be computed with the help of a technique described in [7], using a submanifold of S which is transversal to the leaves of E. Choosing as transversal submanifold the space Q whose elements have the form…”

Section: The Hierarchies Kp (M)mentioning

confidence: 99%

“…Anyhow, to be concrete, and to illustrate more clearly the results, we will restrict ourselves to the case of KP (2) 3 . Indeed, we devote the final section both to provide a hamiltonian interpretation of our previous construction and to identify the KP (2) 3 theory with the Fractional Boussinesq (or sl (2) 3 ) hierarchy described in [1,16,4]. It will be obtained, with its bihamiltonian structure, in the framework of the MarsdenRatiu reduction scheme.…”

Section: Introductionmentioning

confidence: 98%

“…We call those theories KP (m) n . It is our belief that, at least when m and n are coprime, these hierarchies coincide with the type I fractional sl (m) n hierarchies of [16,4]. Anyhow, to be concrete, and to illustrate more clearly the results, we will restrict ourselves to the case of KP (2) 3 .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A note on fractional KdV hierarchies

Casati

Falqui

Magri

et al. 1997

Journal of Mathematical Physics

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We introduce a hierarchy of mutually commuting dynamical systems on a finite number of Laurent series. This hierarchy can be seen as a prolongation of the KP hierarchy, or a "reduction" in which the space coordinate is identified with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV hierarchies are gotten by means of further reductions, obtained by constraining the Laurent series. The case of sl (2) 3 and its bihamiltonian structure is discussed in detail.

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W-Algebras from Soliton Equations and Heisenberg Subalgebras

et al. 1995

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We derive sufficient conditions under which the "second" Hamiltonian structure of a class of generalized KdV-hierarchies defines one of the classical W-algebras obtained through Drinfel'd-Sokolov Hamiltonian reduction. These integrable hierarchies are associated to the Heisenberg subalgebras of an untwisted affine Kac-Moody algebra. When the principal Heisenberg subalgebra is chosen, the well known connection between the Hamiltonian structure of the generalized Drinfel'd-Sokolov hierarchies -the Gel'fand-Dickey algebrasand the W-algebras associated to the Casimir invariants of a Lie algebra is recovered. After carefully discussing the relations between the embeddings of A 1 = sl(2, C) into a simple Lie algebra g and the elements of the Heisenberg subalgebras of g(1) , we identify the class of W-algebras that can be defined in this way. For A n , this class only includes those 1 pousa@gaes.usc.es 2 gallas@gaes.usc.es 3 miramontes@gaes.usc.es 4 joaquin@gaes.usc.es associated to the embeddings labelled by partitions of the form n + 1 = k (m) + q (1) and n + 1 = k (m + 1) + k (m) + q (1). August 1994 1.Introduction.

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Generalized Drinfel'd-Sokolov hierarchies

Cited by 76 publications

References 15 publications

$ \mathcal{W} $ symmetry and integrability of higher spin black holes

$ \mathcal{W} $ symmetry and integrability of higher spin black holes

A note on fractional KdV hierarchies

W-Algebras from Soliton Equations and Heisenberg Subalgebras

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