Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties
- PDF / 563,116 Bytes
- 27 Pages / 439.37 x 666.142 pts Page_size
- 10 Downloads / 201 Views
Mathematische Zeitschrift
Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties Dmitri I. Panyushev1 · Oksana S. Yakimova2 To the memory of Bertram Kostant Received: 7 February 2019 / Accepted: 7 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra g, we obtain several results on the completeness of homogeneous Poisson-commutative subalgebras of S(g) on coadjoint orbits. This concerns, in particular, Gelfand–Tsetlin and Mishchenko–Fomenko subalgebras. Our results reveal the crucial role of nilpotent orbits and sheets in g g∗ . Keywords Integrable systems · Moment map · Coisotropic actions · Coadjoint orbits Mathematics Subject Classification 17B63 · 14L30 · 17B08 · 17B20 · 22E46
Introduction Symplectic manifolds or varieties (M, ω) provide a natural setting for integrable systems. The algebra of “suitable” functions on M, Fun(M), carries a Poisson bracket, and connections with Geometric Representation Theory occur if a Hamiltonian action of a Lie group Q on M is given. Let μ : M → q∗ = (Lie Q)∗ be the corresponding moment mapping and S(q) the symmetric algebra of q. Then S(q) is a Poisson algebra and the co-morphism μ∗ : S(q) → Fun(M) is a Poisson homomorphism. Therefore, if A ⊂ S(q) is Poisson-commutative, then so
The research of the first author was supported by the Russian Foundation for Sciences. The second author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—project number 330450448.
B
Dmitri I. Panyushev [email protected] Oksana S. Yakimova [email protected]
1
Institute for Information Transmission Problems of the R.A.S., Bolshoi Karetnyi per. 19, Moscow 127051, Russia
2
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Cologne, Germany
123
D. I. Panyushev, O. S. Yakimova
is μ∗ (A). For a coisotropic Hamiltonian action of Q on M, one obtains a completely integrable system on M, see [44]. The key point here is the existence of a Poisson-commutative algebra A ⊂ S(q) that is complete, i.e., it provides a complete family in involution on a generic Q-orbit in the image of μ, see Definition 1. Two the most celebrated examples of Poisson-commutative subalgebras are the Gelfand– Tsetlin subalgebras of S(sln ) and S(son ). Their definitions go back to [11,12,15,16]. The success of that construction heavily relies on the existence of chains of coisotropic actions. We prove that both these algebras are complete on every coadjoint orbit. For arbitrary simple Lie algebras g, a large supply of Poisson-commutative subalgebras of S(g) is given by the argument shift method, see below. Our ground field k is algebraically closed and of characteristic 0. Let G be a reductive algebraic group over k with g = Lie G. Poisson-commutative subalgebras of S(g) attract a great deal of attention, because of their relationship to geometric represent
Data Loading...
