Integrable Systems of Classical Mechanics and Lie Algebras

Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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“…Further references I refer to a few standard monographs such as [78], [79] [184], and [190]. Chapter 13.…”

Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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“…and by Γ θ α,β = 1 4 g α+β N α,β + g θα+β N θα,β + g α+θβ N α,θβ + g θα+θβ N θα,θβ (4) respectively. 2 In the first case, it was found that a solution of these constraints exists only for the Lie algebras sl(n, C) of the A-series, whereas in the second case, explicit solutions were found for the complex Grassmannians SU(p + q)/ S(U(p) × U(q)) of the A III-series when |p − q| 1.…”

Section: Introductionmentioning

confidence: 99%

From dynamical to numerical R-matrices: a case study for the Calogero models

Forger¹,

Winterhalder²

2003

Nuclear Physics B

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“…[35]) and higher spins [36]. In particular, the W -algebra related to the principal SL(2) is finite and Abelian and the corresponding Toda system is called Abelian.…”

Section: Introductionmentioning

confidence: 99%

Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

Cacciatori

Cerchiai

Ferrara

et al. 2014

Fortschritte der Physik

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We analyze the polynomial part of the Iwasawa realization of the coset representative of non compact symmetric Riemannian spaces. We start by studying the role of Kostant's principal SU (2) P subalgebra of simple Lie algebras, and how it determines the structure of the nilpotent subalgebras. This allows us to compute the maximal degree of the polynomials for all faithful representations of Lie algebras. In particular the metric coefficients are related to the scalar kinetic terms while the representation of electric and magnetic charges is related to the coupling of scalars to vector field strengths as they appear in the Lagrangian.We consider symmetric scalar manifolds in N−extended supergravity in various space-time dimensions, elucidating various relations with the underlying Jordan algebras and normed Hurwitz algebras. For magic supergravity theories, our results are consistent with the Tits-Satake projection of symmetric spaces and the nilpotency degree turns out to depend only on the space-time dimension of the theory.These results should be helpful within a deeper investigation of the corresponding supergravity theory, e.g. in studying ultraviolet properties of maximal supergravity in various dimensions.

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Integrable Systems of Classical Mechanics and Lie Algebras

Cited by 533 publications

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Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

From dynamical to numerical R-matrices: a case study for the Calogero models

Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

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