부산대학교 도서관

It 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 gl (n , R) . We here show that the quadratic bracket on gl (n , R) , corresponding to the r-matrix defined by the splitting of 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 ∗ 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. [ABSTRACT FROM AUTHOR]