On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups

By Mikhail Kogan, Andrei Zelevinsky. 2002

60 citations recorded by the source corpus.

We provide an explicit description of symplectic leaves of a simply connected connected semisimple complex Lie group equipped with the standard Poisson-Lie structure. This sharpens previously known descriptions of the symplectic leaves as connected components of certain varieties. Our main tool is the machinery of twisted generalized minors. They also allow us to present several quasi-commuting coordinate systems on every symplectic leaf. As a consequence, we construct new completely integrable systems on some special symplectic leaves.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b93354f46b37463f3f0e9f6d6b271c7464104be3

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Citations in the corpus

Introduction to Cluster Algebras (2018, background): “ymplectic manifold whose 2-form is given by ω(ξf,ξg) = {f,g}and thus Mp is called the symplectic leaf of M through p. Example 5.9. The symplectic leaves of GLn have been explicitly computed (e.g. see [35] and references therein). In particular, it is known that each double Bruhat cell in GLn is foli”
Quantum cluster algebras (2004, background): “Not very surprisingly, the skew-symmetric matrix that appears here is the one describing the standard Poisson structure in the double cell in question; this matrix was calculated in[16,11].”
Cluster algebras III: Upper bounds and double Bruhat cells (2003, background): “, they are associated with pairs of elements (u,v) of the Weyl group. Double Bruhat cells make a natural appearance in the context of quantum groups [3], total positivity [14], and integrable systems [10, 11]. The connection between cluster algebras and double Bruhat cells was hinted upon in the pap”
Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective (2009, methodology): “distinguished symplectic leaf and a is an element of the Cartan subgroup, and (ii) the dimension of symplectic leaves in G u,v equals l(u)+l(v)+corank(uv 1 Id), see [ 26 ] and [30].”
Cluster algebras and Weil-Petersson forms (2003, methodology): “m a log-canonical coordinate system with respect to the standard Poisson-Lie structure on the group. Moreover, the corresponding Poisson bracket is compatible with cluster transformations (see [FZ1], [KZ] for deﬁnitions and details). Another example, discussed in [GSV], is the real Grassmannian G(k,”