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 |
How to Cite This Entry:
Toda lattices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toda_lattices&oldid=13616
This article was adapted from an original article by M. Adler (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article