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Journal articles on the topic 'Poisson brackets'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 March 2023

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1

KAHNG, BYUNG-JAY. "DEFORMATION QUANTIZATION OF CERTAIN NONLINEAR POISSON STRUCTURES." International Journal of Mathematics 09, no. 05 (August 1998): 599–621. http://dx.doi.org/10.1142/s0129167x98000269.

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As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain nonlinear Poisson brackets which are "cocycle perturbations" of the linear Poisson bracket. We show that these special Poisson brackets are equivalent to Poisson brackets of central extension type, which resemble the central extensions of an ordinary Lie bracket via Lie algebra cocycles. We are able to formulate (strict) deformation quantizations of these Poisson brackets by means of twisted group C*-algebras. We also indicate that these deformation quantizations can be used to construct some specific non-compact quantum groups.

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2

BATALIN, IGOR, and ROBERT MARNELIUS. "DUALITIES BETWEEN POISSON BRACKETS AND ANTIBRACKETS." International Journal of Modern Physics A 14, no. 32 (December 30, 1999): 5049–73. http://dx.doi.org/10.1142/s0217751x99002384.

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Recently it has been shown that antibrackets may be expressed in terms of Poisson brackets and vice versa for commuting functions in the original bracket. Here we also introduce generalized brackets involving higher antibrackets or higher Poisson brackets where the latter are of a new type. We give generating functions for these brackets for functions in arbitrary involutions in the original bracket. We also give master equations for generalized Maurer–Cartan equations. The presentation is completely symmetric with respect to Poisson brackets and antibrackets.

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3

Bruce, Andrew James. "Odd Jacobi Manifolds and Loday-Poisson Brackets." Journal of Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/630749.

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We construct a nonskew symmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Furthermore, we show that the Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field.

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4

De Sole, Alberto, Victor G. Kac, Daniele Valeri, and Minoru Wakimoto. "Poisson Λ-brackets for Differential–Difference Equations." International Mathematics Research Notices 2020, no. 13 (October 30, 2018): 4144–90. http://dx.doi.org/10.1093/imrn/rny242.

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Abstract We introduce the notion of a multiplicative Poisson $\lambda$-bracket, which plays the same role in the theory of Hamiltonian differential–difference equations as the usual Poisson $\lambda$-bracket plays in the theory of Hamiltonian partial differential equations (PDE). We classify multiplicative Poisson $\lambda$-brackets in one difference variable up to order 5. As an example, we demonstrate how to apply the Lenard–Magri scheme to a compatible pair of multiplicative Poisson $\lambda$-brackets of order 1 and 2, to establish integrability of the Volterra chain.

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5

FERGUSON, JAMES T. "SECOND-ORDER DEFORMATIONS OF HYDRODYNAMIC-TYPE POISSON BRACKETS." Glasgow Mathematical Journal 51, A (February 2009): 75–82. http://dx.doi.org/10.1017/s0017089508004795.

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AbstractThis paper is concerned with the properties of differential-geometric-type Poisson brackets specified by a differential operator of degree 2. It also considers the conditions required for such a Poisson bracket to form a bi-Hamiltonian structure with a hydrodynamic-type Poisson bracket.

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6

Chen, K. C. "Constructing Poisson and Dissipative Brackets of Mixtures by using Lagrangian-to-Eulerian Transformation." Journal of Mechanics 26, no. 2 (June 2010): 219–28. http://dx.doi.org/10.1017/s1727719100003075.

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AbstractThis paper aims to construct the bracket formalism of mixture continua by using the method of Lagrangian- to-Eulerian (LE) transformation. The LE approach first builds up the transformation relations between the Eulerian state variables and the Lagrangian canonical variables, and then transforms the bracket in Lagrangian form to the bracket in Eulerian form. For the conservative part of the bracket formalism, this study systematically generates the noncanonical Poisson brackets of a two-component mixture. For the dissipative part, we deduce the Eulerian-variable-based dissipative brackets for viscous and diffusive mechanisms from their Lagrangian-variable-based counterparts. Finally, the evolution equations of a micromorphic fluid, which can be treated as a multi-component mixture, are derived by constructing its Poisson and dissipative brackets.

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7

Bimonte, G., G. Esposito, G. Marmo, and C. Stornaiolo. "Peierls Brackets in Field Theory." International Journal of Modern Physics A 18, no. 12 (May 10, 2003): 2033–39. http://dx.doi.org/10.1142/s0217751x03015453.

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Peierls brackets are part of the space-time approach to quantum field theory, and provide a Poisson bracket which, being defined for pairs of observables which are group invariant, is group invariant by construction. It is therefore well suited for combining the use of Poisson brackets and the full diffeomorphism group in general relativity. The present paper provides an introduction to the topic, with applications to gauge field theory.

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8

Guttenberg, S. "Derived Brackets from Super-Poisson Brackets." Nuclear Physics B - Proceedings Supplements 171 (September 2007): 279–80. http://dx.doi.org/10.1016/j.nuclphysbps.2007.06.025.

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9

Beltiţă, Daniel, Tomasz Goliński, and Alice-Barbara Tumpach. "Queer Poisson brackets." Journal of Geometry and Physics 132 (October 2018): 358–62. http://dx.doi.org/10.1016/j.geomphys.2018.06.013.

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10

Eastwood, M., and G. Marí Beffa. "Geometric Poisson brackets on Grassmannians and conformal spheres." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 3 (June 2012): 525–61. http://dx.doi.org/10.1017/s0308210510001071.

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We relate the geometric Poisson brackets on the 2-Grassmannian in ℝ4 and on the (2, 2) Möbius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Möbius sphere does not restrict to the space of differential invariants of Schwarzian type. But when the concept of conformal natural frame is transported from the conformal sphere into the Grassmannian, and the Poisson bracket is written in terms of the Grassmannian natural frame, it restricts and results in either a decoupled system or a complexly coupled system of Korteweg–de Vries (KdV) equations, depending on the character of the invariants. We also show that the bi-Hamiltonian Grassmannian geometric brackets are equivalent to the non-commutative KdV bi-Hamiltonian structure. Both integrable systems and Hamiltonian structure can be brought back to the conformal sphere.

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11

Mokhov, O. I. "Dubrovin-Novikov type Poisson brackets (DN-brackets)." Functional Analysis and Its Applications 22, no. 4 (1989): 336–38. http://dx.doi.org/10.1007/bf01077434.

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12

BRATCHIKOV, A. V. "REALIZATIONS OF OBSERVABLES IN HAMILTONIAN SYSTEMS WITH FIRST CLASS CONSTRAINTS." International Journal of Geometric Methods in Modern Physics 04, no. 04 (June 2007): 517–22. http://dx.doi.org/10.1142/s0219887807002168.

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In a Hamiltonian system with first class constraints, observables can be defined as elements of a quotient Poisson bracket algebra. In the gauge fixing method, observables form a quotient Dirac bracket algebra. We show that these two algebras are isomorphic. A new realization of the observable algebras through the original Poisson bracket is found. Generators, brackets and pointwise products of the algebras under consideration are calculated.

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13

Ciaglia, F. M., F. Di Cosmo, A. Ibort, G. Marmo, and L. Schiavone. "Covariant variational evolution and Jacobi brackets: Particles." Modern Physics Letters A 35, no. 23 (May 15, 2020): 2020001. http://dx.doi.org/10.1142/s0217732320200011.

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The formulation of covariant brackets on the space of solutions to a variational problem is analyzed in the framework of contact geometry. It is argued that the Poisson algebra on the space of functionals on fields should be read as a Poisson subalgebra within an algebra of functions equipped with a Jacobi bracket on a suitable contact manifold.

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14

Kolev, Boris. "Poisson brackets in Hydrodynamics." Discrete & Continuous Dynamical Systems - A 19, no. 3 (2007): 555–74. http://dx.doi.org/10.3934/dcds.2007.19.555.

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15

Arthamonov, S. "Modified double Poisson brackets." Journal of Algebra 492 (December 2017): 212–33. http://dx.doi.org/10.1016/j.jalgebra.2017.08.025.

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16

Kisisel, Ali Ulas Özgür. "On quadratic Poisson brackets." Journal of Mathematical Physics 46, no. 4 (April 2005): 042701. http://dx.doi.org/10.1063/1.1866221.

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17

Crainic, Marius, and Rui Loja Fernandes. "Integrability of Poisson Brackets." Journal of Differential Geometry 66, no. 1 (January 2004): 71–137. http://dx.doi.org/10.4310/jdg/1090415030.

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18

Fernandes, Rui Loja, and Philippe Monnier. "Linearization of Poisson Brackets." Letters in Mathematical Physics 69, no. 1-3 (July 2004): 89–114. http://dx.doi.org/10.1007/s11005-004-0340-4.

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19

Alekseev, A., and I. Davydenkova. "Inequalities from Poisson brackets." Indagationes Mathematicae 25, no. 5 (October 2014): 846–71. http://dx.doi.org/10.1016/j.indag.2014.07.003.

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20

Zambrano, B. A. "Poisson brackets on some skew PBW extensions." Algebra and Discrete Mathematics 29, no. 2 (2020): 277–302. http://dx.doi.org/10.12958/adm1037.

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21

Elfimov, Boris M., and Alexey A. Sharapov. "Lie and Leibniz algebras of lower-degree conservation laws." Journal of Physics A: Mathematical and Theoretical 55, no. 6 (January 21, 2022): 065201. http://dx.doi.org/10.1088/1751-8121/ac477d.

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Abstract A relationship between the asymptotic and lower-degree conservation laws in (non-) linear gauge theories is considered. We show that the true algebraic structure underlying asymptotic charges is that of Leibniz rather than Lie. The Leibniz product is defined through the derived bracket construction for the natural Poisson brackets and the BRST differential. Only in particular, though not rare, cases that the Poisson brackets of lower-degree conservation laws vanish modulo central charges, the corresponding Leibniz algebra degenerates into a Lie one. The general construction is illustrated by two standard examples: Yang–Mills theory and Einstein’s gravity.

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22

Lorenzoni, P., and R. Vitolo. "Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds." Journal of Geometry and Physics 149 (March 2020): 103573. http://dx.doi.org/10.1016/j.geomphys.2019.103573.

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23

Abramov, Viktor. "Generalization of Nambu–Hamilton Equation and Extension of Nambu–Poisson Bracket to Superspace." Universe 4, no. 10 (October 15, 2018): 106. http://dx.doi.org/10.3390/universe4100106.

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We propose a generalization of the Nambu–Hamilton equation in superspace R 3 | 2 with three real and two Grassmann coordinates. We construct the even degree vector field in the superspace R 3 | 2 by means of the right-hand sides of the proposed generalization of the Nambu–Hamilton equation and show that this vector field is divergenceless in superspace. Then we show that our generalization of the Nambu–Hamilton equation in superspace leads to a family of ternary brackets of even degree functions defined with the help of a Berezinian. This family of ternary brackets is parametrized by the infinite dimensional group of invertible second order matrices, whose entries are differentiable functions on the space R 3 . We study the structure of the ternary bracket in a more general case of a superspace R n | 2 with n real and two Grassmann coordinates and show that for any invertible second order functional matrix it splits into the sum of two ternary brackets, where one is the usual Nambu–Poisson bracket, extended in a natural way to even degree functions in a superspace R n | 2 , and the second is a new ternary bracket, which we call the Ψ -bracket, where Ψ can be identified with an invertible second order functional matrix. We prove that the ternary Ψ -bracket as well as the whole ternary bracket (the sum of the Ψ -bracket with the usual Nambu–Poisson bracket) is totally skew-symmetric, and satisfies the Leibniz rule and the Filippov–Jacobi identity ( Fundamental Identity).

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24

BIMONTE, G., P. SALOMONSON, A. SIMONI, and A. STERN. "POISSON BRACKET ALGEBRA FOR CHIRAL GROUP ELEMENTS IN THE WZNW MODEL." International Journal of Modern Physics A 07, no. 24 (September 30, 1992): 6159–74. http://dx.doi.org/10.1142/s0217751x92002799.

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We examine the Wess-Zumino-Novikov-Witten (WZNW) model on a circle and compute the Poisson bracket algebra for left- and right-moving chiral group elements. Our computations apply for arbitrary groups and arbitrary boundary conditions, the latter being characterized by the monodromy matrix. Unlike in previous treatments, the Poisson brackets do not require specifying a particular parametrization of the group valued fields in terms of angles spanning the group. We do however find it necessary to make a gauge choice, as the chiral group elements are not gauge invariant observables. (On the other hand, the quadratic form of the Poisson brackets may be defined independently of a gauge fixing.) Gauge invariant observables can be formed from the monodromy matrix and these observbles are seen to commute in the quantum theory.

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25

Ibáñez, Raúl, Manuel de León, Juan C. Marrero, and David Martı́n de Diego. "Dynamics of generalized Poisson and Nambu–Poisson brackets." Journal of Mathematical Physics 38, no. 5 (May 1997): 2332–44. http://dx.doi.org/10.1063/1.531960.

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26

Okubo, Susumu. "Poisson Brackets and Nijenhuis Tensor." Zeitschrift für Naturforschung A 52, no. 1-2 (February 1, 1997): 76–78. http://dx.doi.org/10.1515/zna-1997-1-220.

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Abstract Many integrable models satisfy the zero Nijenhuis tensor condition. Although its application for discrete systems is then straightforward, there exist some complications to utilize the condition for continuous infinite dimensional models. A brief sketch of how we deal with the problem is explained with an application to a continuous Toda lattice.

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27

Ahn, Jaehyun, Sei-Qwon Oh, and Sujin Park. "POISSON BRACKETS DETERMINED BY JACOBIANS." Journal of the Chungcheong Mathematical Society 26, no. 2 (May 15, 2013): 357–65. http://dx.doi.org/10.14403/jcms.2013.26.2.357.

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28

Lemos, Nivaldo A. "Jacobi’s identity for poisson brackets." American Journal of Physics 68, no. 9 (September 2000): 788. http://dx.doi.org/10.1119/1.1305742.

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29

Mokhov, O. I. "Local third-order Poisson brackets." Russian Mathematical Surveys 40, no. 5 (October 31, 1985): 233–34. http://dx.doi.org/10.1070/rm1985v040n05abeh003689.

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30

Sokolov, V. V., P. A. Eminov, and K. N. Fotov. "Poisson brackets method in ferrohydrodynamics." Physics Procedia 9 (2010): 131–36. http://dx.doi.org/10.1016/j.phpro.2010.11.031.

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31

Buhovsky, Lev, Michael Entov, and Leonid Polterovich. "Poisson brackets and symplectic invariants." Selecta Mathematica 18, no. 1 (September 24, 2011): 89–157. http://dx.doi.org/10.1007/s00029-011-0068-9.

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32

Cantero, María José, and Barry Simon. "Poisson brackets of orthogonal polynomials." Journal of Approximation Theory 158, no. 1 (May 2009): 3–48. http://dx.doi.org/10.1016/j.jat.2007.07.001.

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33

Bering, K. "Family of boundary Poisson brackets." Physics Letters B 486, no. 3-4 (August 2000): 426–30. http://dx.doi.org/10.1016/s0370-2693(00)00778-4.

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34

DUBROVIN, B. A., M. GIORDANO, G. MARMO, and A. SIMONI. "POISSON BRACKETS ON PRESYMPLECTIC MANIFOLDS." International Journal of Modern Physics A 08, no. 21 (August 20, 1993): 3747–71. http://dx.doi.org/10.1142/s0217751x93001521.

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The problem of defining Poisson brackets for a degenerate Lagrangian without introduction of canonical variables is discussed. More generally, we introduce and give a complete geometrical description of a class of Poisson brackets on a presymplectic manifold. The construction is illustrated by both finite-dimensional and field-theoretic examples.

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35

Tezuka, K. I. "Poisson brackets, strings and membranes." European Physical Journal C 25, no. 3 (October 2002): 465–68. http://dx.doi.org/10.1007/s10052-002-1006-y.

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36

Damianou, Pantelis A., and Fani Petalidou. "Poisson Brackets with Prescribed Casimirs." Canadian Journal of Mathematics 64, no. 5 (October 1, 2012): 991–1018. http://dx.doi.org/10.4153/cjm-2011-082-2.

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Abstract We consider the problem of constructing Poisson brackets on smooth manifolds M with prescribed Casimir functions. If M is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on M, while, in the case where M is of odd dimension, our objective is achieved using a convenient almost cosymplectic structure. Several examples and applications are presented.

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37

Polishchuk, A. "Algebraic geometry of Poisson brackets." Journal of Mathematical Sciences 84, no. 5 (May 1997): 1413–44. http://dx.doi.org/10.1007/bf02399197.

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38

Szasz A., Vincze Gy. "Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator." JOURNAL OF ADVANCES IN PHYSICS 4, no. 1 (March 22, 2014): 404–26. http://dx.doi.org/10.24297/jap.v4i1.6966.

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Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

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39

Barbaresco, Frédéric. "Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation." Entropy 24, no. 11 (November 9, 2022): 1626. http://dx.doi.org/10.3390/e24111626.

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The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau’s cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain’s PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain’s works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie–Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation

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40

Melani, Valerio. "Poisson bivectors and Poisson brackets on affine derived stacks." Advances in Mathematics 288 (January 2016): 1097–120. http://dx.doi.org/10.1016/j.aim.2015.11.008.

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41

Tsiganov, A. V. "A family of the Poisson brackets compatible with the Sklyanin bracket." Journal of Physics A: Mathematical and Theoretical 40, no. 18 (April 17, 2007): 4803–16. http://dx.doi.org/10.1088/1751-8113/40/18/008.

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42

Grabowski, Janusz. "Z-Graded Extensions of Poisson Brackets." Reviews in Mathematical Physics 09, no. 01 (January 1997): 1–27. http://dx.doi.org/10.1142/s0129055x97000026.

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A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an 'integral' of the Koszul–Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dμ, dν]P=d{μ, ν}P. A naturally defined generalized Hamiltonian map is proved to be a homomorphism between { , }P and the Frölicher–Nijenhuis bracket of vector valued forms. Also relations of this graded Poisson bracket to the Schouten–Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an 'integral' of the Schouten–Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.

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43

Zhang, Pumei. "Algebraic properties of compatible Poisson brackets." Regular and Chaotic Dynamics 19, no. 3 (May 2014): 267–88. http://dx.doi.org/10.1134/s1560354714030010.

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44

Damianou, Pantelis A. "Poisson Brackets after Jacobi and Plücker." Regular and Chaotic Dynamics 23, no. 6 (November 2018): 720–34. http://dx.doi.org/10.1134/s1560354718060072.

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45

Ferapontov, E. V. "Compatible Poisson brackets of hydrodynamic type." Journal of Physics A: Mathematical and General 34, no. 11 (March 14, 2001): 2377–88. http://dx.doi.org/10.1088/0305-4470/34/11/328.

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46

Acatrinei, Ciprian Sorin. "Dimensional reduction for generalized Poisson brackets." Journal of Mathematical Physics 49, no. 2 (February 2008): 022903. http://dx.doi.org/10.1063/1.2863440.

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47

Valtancoli, P. "Path integral and noncommutative Poisson brackets." Journal of Mathematical Physics 56, no. 6 (June 2015): 063501. http://dx.doi.org/10.1063/1.4922107.

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48

Marsden, J. E., R. Montgomery, P. J. Morrison, and W. B. Thompson. "Covariant poisson brackets for classical fields." Annals of Physics 169, no. 1 (June 1986): 29–47. http://dx.doi.org/10.1016/0003-4916(86)90157-0.

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49

Naef, Florian. "Poisson brackets in Kontsevich’s “Lie World”." Journal of Geometry and Physics 155 (September 2020): 103741. http://dx.doi.org/10.1016/j.geomphys.2020.103741.

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50

Yan, C. C. "Quantization based on generalized Poisson brackets." Il Nuovo Cimento B Series 11 107, no. 11 (November 1992): 1239–59. http://dx.doi.org/10.1007/bf02726090.

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