a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which...

derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra...

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...

by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu–Poisson manifolds with more than one Hamiltonian...

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...

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...

integrable. Symplectic manifolds are special cases of a Poisson manifold. A multisymplectic manifold of degree k is a manifold equipped with a closed...

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...

almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost...

superalgebra. Every symplectic supermanifold is a Poisson supermanifold but not vice versa. Poisson manifold Poisson algebra Noncommutative geometry v t e...

Poisson manifold there is a Lie algebroid structure on A ∗ {\displaystyle A^{*}} induced by this Poisson structure. Analogous to the Poisson manifold...

Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to...

Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation...

where the phase space is a symplectic manifold, or possibly a Poisson manifold. Related structures include the Poisson–Lie groups and Kac–Moody algebras....

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...

tensor expressions. In certain specialized applications, such as on Poisson manifolds, the relationship may fail to be an isomorphism at singular points...

arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to...

gives altogether four different classes of constraints. Consider a Poisson manifold M with a smooth Hamiltonian over it (for field theories, M would be...

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...

his results is a formal deformation quantization that holds for any Poisson manifold. He also introduced the Kontsevich integral, a topological invariant...

Symplectic manifold Symplectic structure Symplectomorphism Contact structure Contact geometry Hamiltonian system Sasakian manifold Poisson manifold Möbius...

consider a Poisson manifold instead of a symplectic space for the classical theory and perform an ħ-deformation of the corresponding Poisson algebra or...

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...

limit Phase space Symplectic manifold Liouville's theorem (Hamiltonian) Poisson bracket Poisson algebra Poisson manifold Antibracket algebra Hamiltonian...

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...

the point, much like Weinstein's theorem for the local structure of Poisson manifolds. The remaining question of the local structure is: what does a generalized...

contributions to the study of Dirac manifolds, which generalize both symplectic manifolds and Poisson manifolds, and are related to the Dirac theory...

algebra. The differential forms on a Poisson manifold form a Gerstenhaber algebra. The multivector fields on a manifold form a Gerstenhaber algebra using...

{\displaystyle \mathbb {R} ^{2n}} , equipped with its Poisson bracket (with a generalization to symplectic manifolds, described below). It is a special case of the...

{\displaystyle T^{*}M} associated to Poisson manifolds ( M , π ) {\displaystyle (M,\pi )} are transitive if and only if the Poisson structure π {\displaystyle \pi...