Abstract
AbstractIt is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic r-matrix Poisson brackets on the associative algebra $$\mathrm {gl}(n,{\mathbb {R}})$$
gl
(
n
,
R
)
. We here show that the quadratic bracket on $$\mathrm {gl}(n,{\mathbb {R}})$$
gl
(
n
,
R
)
, corresponding to the r-matrix defined by the splitting of $$\mathrm {gl}(n,{\mathbb {R}})$$
gl
(
n
,
R
)
into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle $$T^* \mathrm {GL}(n,{\mathbb {R}})$$
T
∗
GL
(
n
,
R
)
. This complements the interpretation of the linear r-matrix bracket as a reduction of the canonical Poisson bracket of the cotangent bundle.
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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