Abstract
A Poisson bracket on a manifold M generates standard Lie algebroid structure on the cotangent bundle \(T^{\textstyle *}\!M\). Under some assumptions, this structure can be written as a linear combination of two different algebroid structures. By specifying vector fields X and Y on a given manifold, it is possible to construct a family of Lie algebroids determined by these vector fields. Specializing the Lie algebroid to the case of a Lie algebra on a linear space V , upon defining a pair (F, v), where F ∈ End(V ), and v is an eigenvector of the endomorphism F, allows to generalize Lie bracket given by vector fields. Linear combinations of generalized brackets, under some assumptions, are again Lie brackets. Constructions for some compatible structures are shown, e.g., on a set of antisymmetric matrices related to commutator defined by symmetric matrix.
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The second author G.J. was partially supported by National Science Center, Poland project 2020/01/Y/ST1/00123.
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Dobrogowska, A., Jakimowicz, G., Szajewska, M. (2023). On Some Structures of Lie Algebroids on the Cotangent Bundles. In: Kielanowski, P., Dobrogowska, A., Goldin, G.A., Goliński, T. (eds) Geometric Methods in Physics XXXIX. WGMP 2022. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-30284-8_16
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