Letters in Mathematical Physics

, Volume 104, Issue 8, pp 1045–1052 | Cite as

Explicit Semi-invariants and Integrals of the Full Symmetric \({\mathfrak{sl}_n}\) Toda Lattice

  • Yury B. Chernyakov
  • Alexander S. Sorin


We show how to construct semi-invariants and integrals of the full symmetric \({\mathfrak{sl}_n}\) Toda lattice for all n. Using the Toda equations for the Lax eigenvector matrix we prove the existence of semi-invariants which are homogeneous coordinates in the corresponding projective spaces. Then we use these semi-invariants to construct the integrals. The existence of additional integrals which constitute a full set of independent non-involutive integrals was known but the chopping and Kostant procedures have crucial computational complexities already for low-rank Lax matrices and are practically not applicable for higher ranks. Our new approach solves this problem and results in simple explicit formulae for the full set of independent semi-invariants and integrals expressed in terms of the Lax matrix and its eigenvectors, and of eigenvalue matrices for the full symmetric \({\mathfrak{sl}_n}\) Toda lattice.

Mathematics Subject Classification

37J35 37K10 70H06 82B20 


Liouville integrability integrals of motion semi-invariants full symmetric Toda Lattice flag space non-commutative integrability Lax representation 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Bogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchDubnaRussia
  3. 3.Bogoliubov Laboratory of Theoretical Physics and Veksler and Baldin Laboratory of High Energy PhysicsJoint Institute for Nuclear ResearchDubnaRussia

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