Separation Coordinates, Moduli Spaces and Stasheff Polytopes

Schöbel, Konrad; Веселов, А. П.

doi:10.1007/s00220-015-2332-x

Cited by 14 publications

(16 citation statements)

References 26 publications

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“…These observations suggests a relation between separation coordinates on spheres and moduli spaces of stable curves of genus zero with marked points [12,24,25]. Indeed, our thorough analysis of the case S 3 presented here has led to the following generalisation of Theorem 1.19 [42]. Theorem 1.20.…”

mentioning

confidence: 69%

The Variety of Integrable Killing Tensors on the 3-Sphere

Schöbel¹

2014

SIGMA

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Abstract. Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere S 3 and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all Stäckel systems in these varieties. This allows us to recover the known list of separation coordinates on S 3 in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron K 4 .

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confidence: 69%

The Variety of Integrable Killing Tensors on the 3-Sphere

Schöbel¹

2014

SIGMA

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“…It turns out that this operad restricts to orthogonal separable coordinate systems in normal form, which gives the translation of the mosaic operad fromM 0,n+1 (R) to S 0 (S n−1 ). Moreover, the mosaic operad can also be expressed in a simple way in terms of Stäckel systems [30]. The mosaic operad induces an operad on rooted planar trees, whose composition T • (T 1 , .…”

Section: Operad Structurementioning

confidence: 99%

Are Orthogonal Separable Coordinates Really Classified?

Schöbel¹

2016

SIGMA

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Abstract. We prove that the set of orthogonal separable coordinates on an arbitrary (pseudo-)Riemannian manifold carries a natural structure of a projective variety, equipped with an action of the isometry group. This leads us to propose a new, algebraic geometric approach to the classification of orthogonal separable coordinates by studying the structure of this variety. We give an example where this approach reveals unexpected structure in the well known classification and pose a number of problems arising naturally in this context.

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“…His results were generalized by E. Kalnins, W. Miller [23,24] and S. Benenti [2,3]. A modern discussion of this construction together with an exhaustive list of references may be found in [32].…”

Section: Introductionmentioning

confidence: 97%

“…As a sequence, the Nijenhuis and Haantjes tensors appear in mathematics, mathematical physics and classical mechanics, however, the overwhelming majority of applications is only related with the simple deformations at N = 0 or H = 0, see [10,19,27,28,32,35] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Killing tensors with nonvanishing Haantjes torsion and integrable systems

Tsiganov

2015

Regul. Chaot. Dyn.

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The second-order integrable Killing tensor with simple eigenvalues and vanishing Haantjes torsion is the key ingredient in construction of Liouville integrable systems of Stäckel type. We present two examples of the integrable systems on three-dimensional Euclidean space associated with the second-order Killing tensors possessing nontrivial torsion. Integrals of motion for these integrable systems are the second-and fourth-order polynomials in momenta, which are constructed using a special family of the Killing tensors. may be used as canonical coordinates for K in the domain where they are real, distinct, and functionally independent. In this domainand any distribution spanned by the n − 1 eigenvectors X i is completely integrable. Here N denotes the skew-symmetric (1, 2) Nijenhuis torsion tensor field associated with Kwhere X and Y are arbitrary vector fields and [. , .] denotes the commutator of two vector fields. This construction of the integrable distributions as orthogonal complements of each eigenvector field X i was slightly modified by Nijenhuis [29] and Haantjes [21] for the case where they enable K to possess torsion, but in a controlled manner. In this case, condition (1.1) is replaced by the weaker conditionwhere H is the skew-symmetric (1, 2) Haantjes tensor associated with K(1.2)The vanishing of the Haantjes tensor H is equivalent to the condition that the eigenvector fields of K commute after a reparameterization, and equivalent to the condition that any collection of n − 1 eigenvector fields is integrable. The algebraic relations between the Nijenhuis and Haantjes tensors are discussed in [10].

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Separation Coordinates, Moduli Spaces and Stasheff Polytopes

Cited by 14 publications

References 26 publications

The Variety of Integrable Killing Tensors on the 3-Sphere

The Variety of Integrable Killing Tensors on the 3-Sphere

Are Orthogonal Separable Coordinates Really Classified?

Killing tensors with nonvanishing Haantjes torsion and integrable systems

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