Geodesic
On Extensions, Lie-Poisson Systems, and Dissipation
Journal of Lie theory, Tome 32 (2022) no. 2, pp. 327-382
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Lie-Poisson systems on the dual spaces of unified products are studied. Having been equipped with a twisted 2-cocycle term, the extending structure framework allows not only to study the dynamics on 2-cocycle extensions, but also to (de)couple mutually interacting Lie-Poisson systems. On the other hand, symmetric brackets; such as the double bracket, the Cartan-Killing bracket, the Casimir dissipation bracket, and the Hamilton dissipation bracket are worked out in detail. Accordingly, the collective motion of two mutually interacting irreversible dynamics, as well as the mutually interacting metriplectic flows, are obtained. The theoretical results are illustrated in three examples. As an infinite-dimensional physical model, decompositions of the BBGKY hierarchy are presented. As for the finite-dimensional examples, the coupling of two Heisenberg algebras, and the coupling of two copies of $3D$ dynamics are studied.
Classification : 53D17, 37J37
Mots-clés : Lie-Poisson equation, metriplectic system, unified product 
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     author = {O. Esen and G. Oezcan and S. Suetlue},
     title = {On {Extensions,} {Lie-Poisson} {Systems,} and {Dissipation}},
     journal = {Journal of Lie theory},
     pages = {327--382},
     year = {2022},
     volume = {32},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JLT_2022_32_2_JLT_2022_32_2_a2/}
}
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O. Esen; G. Oezcan; S. Suetlue. On Extensions, Lie-Poisson Systems, and Dissipation. Journal of Lie theory, Tome 32 (2022) no. 2, pp. 327-382. http://geodesic.mathdoc.fr/item/JLT_2022_32_2_JLT_2022_32_2_a2/