The toda flow on a generic orbit is integrable

Communications on Pure and Applied Mathematics, 1999

2015

Explicit Construction of First Integrals for the Toda Flow on a Classical Simple Lie Algebra

Boletim da Sociedade Paranaense de Matemática, 2019

The aim of this paper is to classify invariant flows on Lie group $G$ whose Lie algebra $\mathfrak{g}$ is associative or semisimple. Specifically, we present this classification from the hyperbolicity of the lift flows on $G \times \mathfrak{g}$. Then we apply this construction to some special cases as ${\rm Gl}(2,{\Bbb R})$ and affine Lie group.

2005

This paper shows that the left-invariant geodesic flow on the symplectic group relative to the Frobenius metric is an integrable system that is not contained in the Mishchenko-Fomenko class of rigid body metrics. This system may be expressed as a flow on symmetric matrices and is bi-Hamiltonian. This analysis is extended to cover flows on symmetric matrices when an isomorphism * Research partially supported by the NSF. † Research partially supported by the California Institute of Technology and NSF. ‡ Research partially supported by the Swiss NSF. 1 Introduction 2 with the symplectic Lie algebra does not hold. The two Poisson structures associated with this system, including an analysis of its Casimirs, are completely analyzed. Since the system integrals are not generated by its Casimirs it is shown that the nature of integrability is fundamentally different from that exhibited in the Mischenko-Fomenko setting.

2009

For a given skew symmetric real n × n matrix N , the bracket [X, Y ]N = XN Y − Y N X defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian systemẊ = [X 2 , N ], or equivalently in Lax form, the equatioṅ X = [X, XN + N X] on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure. If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [•, •]) is Lie algebra isomorphic with the symplectic Lie algebra sp(n, N −1) associated to the symplectic form on R n given by N −1. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N −1). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible "frozen bracket" structure. The Poisson diffeomorphism from Sym(n, N) to sp(n, N −1) maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with sp(n, N −1), our system becomes a Mischenko-Fomenko system directly on Sym(n, N). If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that the flow does not linearize on the Jacobian of the spectral curve. Contents 1 Introduction 2 2 The Lie Algebra and the Euler-Poincaré Form 4 3 Poisson Structures 7 4 The Sectional Operator Equations and Relation to Mischenko-Fomenko Flows 15

Journal of Geometry and Physics, 1988

The functional integral for the quantization of the coadjoint orbits of the unitary and orthogonal groups is given by means of an explicit construction of the corresponding~Darbouiø> variables.

Analysis and Mathematical Physics, 2018

Pseudo-H-type groups G r ,s form a class of step-two nilpotent Lie groups with a natural pseudo-Riemannian metric. In this paper the question of complete integrability in the sense of Liouville is studied for the corresponding (pseudo-)Riemannian geodesic flow. Via the isometry group of G r ,s families of first integrals are constructed. A modification of these functions gives a set of dim G r ,s functionally independent smooth first integrals in involution. The existence of a lattice L in G r ,s is guaranteed by recent work of K. Furutani and I. Markina. The complete integrability of the pseudo-Riemannian geodesic flow of the compact nilmanifold L\G r ,s is proved under additional assumptions on the group G r ,s .

2013

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Advances in Mathematics, 2021

This paper first introduces the notion of a Rota-Baxter operator (of weight 1) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the decompositions of Iwasawa and Langlands, carry natural Rota-Baxter operators. Formal inverse of the Rota-Baxter operator on a Lie group is precisely the crossed homomorphism on the Lie group, whose tangent map is the differential operator of weight 1 on a Lie algebra. A factorization theorem of Rota-Baxter Lie groups is proved, deriving directly on the Lie group level, the well-known global factorization theorems of Semenov-Tian-Shansky in his study of integrable systems. As geometrization, the notions of Rota-Baxter Lie algebroids and Rota-Baxter Lie groupoids are introduced, with the former a differentiation of the latter. Further, a Rota-Baxter Lie algebroid naturally gives rise to a post-Lie algebroid, generalizing the well-known fact for Rota-Baxter Lie algebras and post-Lie algebras. It is shown that the geometrization of a Rota-Baxter Lie algebra or a Rota-Baxter Lie group can be realized by its action on a manifold. Examples and applications are provided for these new notions.

2004

Based on the structure of Casimir elements associated with general Hopf algebras there are constructed Liouville-Arnold integrable flows related with naturally induced Poisson structures on arbitrary co-algebra and their deformations. Some interesting special cases including the oscillatory Heisenberg-Weil algebra related co-algebra structures and adjoint with them integrable Hamiltonian systems are considered.