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Mathematics > Differential Geometry
Title: Generalized double bracket vector fields
(Submitted on 4 Apr 2024 (v1), last revised 6 May 2024 (this version, v2))
Abstract: By introducing a metriplectic tensor, we generalize the double bracket vector fields defined on semi-simple Lie algebras to the case of Poisson manifolds endowed with a pseudo-Riemannian metric. We also construct a generalization of the normal metric on an adjoint orbit of a compact semi-simple Lie algebra such that the above vector fields, when restricted to a symplectic leaf, become gradient vector fields. We illustrate the newly introduced objects to a variety of examples and carefully discuss complications that arise when the pseudo-Riemannian metric does not induce a non-degenerate metric on parts of the symplectic leaves.
Submission history
From: Zohreh Ravanpak [view email][v1] Thu, 4 Apr 2024 06:08:05 GMT (21993kb,D)
[v2] Mon, 6 May 2024 18:11:52 GMT (15287kb,D)
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