Periodic Toda Lattices Associated to Cartan Matrices

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, j ≤ n. For a mechanical interpretation, consider n unit mass particles on a circle that are connected by exponential springs. In [33], 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:

$$\begin{array}{*{20}{c}} {{V_{al}} = {V_1} + \exp ({q_{l + 1}} - {q_1}),l \geqslant 2,} \\ {{V_{bl}} = {V_{1 - 1}} + \exp ({q_1}) + \exp ( - {q_1} - {q_2}),l \geqslant 2,} \\ {{V_{cl}} = {V_{1 - 1}} + \exp (2{q_1}) + \exp ( - 2{q_1}),l \geqslant 3,} \\ {{V_{dl}} = {V_{1 - 1}} + \exp ({q_{l - 1}} + {q_1}) + \exp ( - {q_1} - {q_2}),l \geqslant 4.}\end{array}$$