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# Periodic Toda Lattices Associated to Cartan Matrices

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Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 47)

## Abstract

In its original form, the n-particle periodic Toda lattice is given by the Hamiltonian on R 2n
$$H = \frac{1}{2}\sum\limits_{i = 1}^n {p_i^2} + \sum\limits_{i = 1}^n {{e^{{q_i} - {q_{i + 1}}}}} ,$$
where q n+1 = q 1; the symplectic structure is the canonical one, {q i , q j } = {p i , p j } = 0 and {q i , p j } = δ ij , where 1 ≤ i, jn. For a mechanical interpretation, consider n unit mass particles on a circle that are connected by exponential springs. In , Bogoyavlensky proposed a Lie algebraic generalization, where the original Toda lattice corresponds to the root system a n−1. Denoting by l the rank of the root system, the general form of the Hamiltonian is
$$H = \frac{1}{2}\sum\limits_{i = 1}^n {p_i^2} + V \bullet ,$$
where qn = l + 1 for the root systems a l , e6, e7, g2 and n = l for the other root systems. Denoting
$${V_k}: = \sum\limits_{i = 1}^k {{e^{{q_i} - {q_{i + 1}}}}}$$
the potential V is given for the root systems that correspond to the classical Lie algebras by the following expressions:
$$\begin{array}{*{20}{c}} {{V_{al}} = {V_1} + \exp ({q_{l + 1}} - {q_1}),l \geqslant 2,} \\ {{V_{bl}} = {V_{1 - 1}} + \exp ({q_1}) + \exp ( - {q_1} - {q_2}),l \geqslant 2,} \\ {{V_{cl}} = {V_{1 - 1}} + \exp (2{q_1}) + \exp ( - 2{q_1}),l \geqslant 3,} \\ {{V_{dl}} = {V_{1 - 1}} + \exp ({q_{l - 1}} + {q_1}) + \exp ( - {q_1} - {q_2}),l \geqslant 4.}\end{array}$$

## Keywords

Irreducible Component Simple Root Abelian Variety Dynkin Diagram Null Vector
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2004

## Authors and Affiliations

1. 1.Department of MathematicsBrandeis UniversityWalthamUSA
2. 2.Department of MathematicsUniversity of LouvainLouvain-la-NeuveBelgium
3. 3.Laboratoire de Mathématiques et ApplicationsUniversité de PoitiersFuturoscopeFrance