A new look at completely integrable systems and double Lie groups

1998 ◽  
pp. 159-172 ◽  
Author(s):  
G. Marmo ◽  
A. Ibort
Keyword(s):  
Lie Groups ◽  
Integrable Systems ◽  
Completely Integrable Systems ◽  
Completely Integrable ◽  
New Look

scholarly journals Completely Integrable Systems — A Generalization

1997 ◽  
Vol 12 (22) ◽  
pp. 1637-1648 ◽  
Author(s):  
D. V. Alekseevsky ◽  
J. Grabowski ◽  
G. Marmo ◽  
P. W. Michor
Keyword(s):  
Dynamical Systems ◽  
Lie Groups ◽  
Integrable Systems ◽  
Completely Integrable Systems ◽  
Completely Integrable ◽  
Slight Generalization

We present a slight generalization of the notion of completely integrable systems such that they can be integrated by quadratures. We use this generalization to integrate dynamical systems on double Lie groups.

scholarly journals Transverse Hilbert schemes and completely integrable systems

2017 ◽  
Vol 4 (1) ◽  
pp. 263-272 ◽  
Author(s):  
Niccolò Lora Lamia Donin
Keyword(s):  
Integrable Systems ◽  
Hilbert Scheme ◽  
Symplectic Form ◽  
Initial Surface ◽  
Hilbert Schemes ◽  
Completely Integrable Systems ◽  
Completely Integrable ◽  
Symplectic Surface ◽  
Complex Symplectic ◽  
Minimum Polynomial

Abstract In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].

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