a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which...
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more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There...
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derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra...
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Nambu mechanics (redirect from Nambu-Poisson manifold)
by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu–Poisson manifolds with more than one Hamiltonian...
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of functions. If the odd Poisson bi-vector π i j {\displaystyle \pi ^{ij}} is invertible, one has an odd symplectic manifold. In that case, there exists...
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mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure...
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Hamiltonian vector field (section Poisson bracket)
on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions f and g on the manifold is itself a Hamiltonian...
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Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation...
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Geometric quantization (section Poisson manifolds)
be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's...
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of a Poisson manifold. A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form. A polysymplectic manifold is a...
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Poisson manifold there is a Lie algebroid structure on A ∗ {\displaystyle A^{*}} induced by this Poisson structure. Analogous to the Poisson manifold...
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superalgebra. Every symplectic supermanifold is a Poisson supermanifold but not vice versa. Poisson manifold Poisson algebra Noncommutative geometry v t e...
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that defines the Poisson bracket. A closely related type of manifold is a contact manifold. A combinatorial manifold is a kind of manifold which is discretization...
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almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost...
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limit Phase space Symplectic manifold Liouville's theorem (Hamiltonian) Poisson bracket Poisson algebra Poisson manifold Antibracket algebra Hamiltonian...
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arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to...
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his results is a formal deformation quantization that holds for any Poisson manifold. He also introduced the Kontsevich integral, a topological invariant...
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Poisson boundary Poisson bracket, see Hamiltonian mechanics header Poisson games Poisson manifold Poisson ring Poisson supermanifold Poisson–Charlier polynomials...
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be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's...
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Daniel Sternheimer, Lichnerowicz formulated the first definitions of a Poisson manifold in terms of a bivector, the counterpart of a (symplectic) differential...
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Symplectic manifold Symplectic structure Symplectomorphism Contact structure Contact geometry Hamiltonian system Sasakian manifold Poisson manifold Möbius...
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where the phase space is a symplectic manifold, or possibly a Poisson manifold. Related structures include the Poisson–Lie groups and Kac–Moody algebras....
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First-class constraint (section Poisson brackets)
gives altogether four different classes of constraints. Consider a Poisson manifold M with a smooth Hamiltonian over it (for field theories, M would be...
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Novikov ring Poisson 1. 2. Poisson algebra. 3. A Poisson manifold generalizes a symplectic manifold. 4. A Poisson–Lie group, a Poisson manifold that also...
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Musical isomorphism (category Riemannian manifolds)
lowering of indices. In certain specialized applications, such as on Poisson manifolds, the relationship may fail to be an isomorphism at singular points...
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algebra. The differential forms on a Poisson manifold form a Gerstenhaber algebra. The multivector fields on a manifold form a Gerstenhaber algebra using...
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coordinates in which the Poisson bivector is constant (plain flat Poisson brackets). For the general formula on arbitrary Poisson manifolds, cf. the Kontsevich...
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algebra of operators on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit, and properties of the non-commutative...
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Moyal product (section On manifolds)
{\displaystyle \mathbb {R} ^{2n}} , equipped with its Poisson bracket (with a generalization to symplectic manifolds, described below). It is a special case of the...
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Hamiltonian mechanics (section Poisson algebras)
The symplectic structure induces a Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra. If...
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