Toda lattices

There are many Toda systems spawned by Toda's nearest neighbour linking of anharmonic oscillators on the line [a1]. A convenient container is the -Toda system, first introduced and studied comprehensively in [a2]; see also [a3].

Let be a bi-infinite or semi-infinite matrix flowing as follows (, the shift operator):

, with Borel decomposition

where and are lower triagonal and .

Define

then

; , with eigenvectors (, ):

; .

Let

the crucial identity

is equivalent to the bilinear identities for the tau-functions

which characterize the solution.

The -Toda system (which can always be imbedded in the -Toda system) is just the -flow for , i.e. it just involves ignoring and in effect freezing at one value. This is equivalent to the Grassmannian flag , , where

or, alternatively, it is characterized by the left-hand side of the bilinear identities for and frozen (or suppressed). The semi-infinite ( or ) Toda system involves setting , , and , in which case and are polynomials in of degree at most .

The famous triagonal Toda system — the original Toda system — is equivalent to the reduction or, equivalently, or, equivalently, . In general, the -gonal Toda system is equivalent to or, equivalently,

The -periodic -Toda system is a -Toda lattice such that . One can of course consider more than one reduction at a time. For example, the -periodic triagonal Toda lattice [a4] linearizes on the Jacobian of a hyper-elliptic curve (the associated spectral curve) with the being essentially theta-functions where in , , the flat coordinates on .

One can also consider in this context Toda flows going with different Lie algebras:

where , , with , being the Cartan matrix of Kac–Moody Lie algebras by extended Dynkin diagrams (cf. also Kac–Moody algebra). The non-periodic case involves being the Cartan matrix of a simple Lie algebra, in which case . The former case linearizes on Abelian varieties [a4] and the latter on "non-compact" Abelian varieties [a5].

References

[a1]	M. Toda, "Vibration of a chain with a non-linear interaction" J. Phys. Soc. Japan , 22 (1967) pp. 431–436
[a2]	K. Ueno, K. Takasaki, "Toda lattice hierarchy" Adv. Studies Pure Math. , 4 (1984) pp. 1–95
[a3]	M. Adler, P. van Moerbeke, "Group factorization, moment matrices and Toda latices" Internat. Math. Research Notices , 12 (1997)
[a4]	M. Adler, P. van Moerbeke, "Completely integrable systems, Euclidean Lie algebras and curves; Linearization of Hamiltonians systems, Jacoby varieties and representation theory" Adv. Math. , 38 (1980) pp. 267–379
[a5]	B. Konstant, "The solution to a generalized Toda lattice and representation theory" Adv. Math. , 34 (1979) pp. 195–338