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

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...

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

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

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

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...

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...

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...

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

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

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

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

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

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...

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

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...

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

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

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

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

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

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

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...

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

the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, including integrable systems, Poisson geometry...

Retrieved 28 April 2021. Nasar, Sylvia; Gruber, David (21 August 2006). "Manifold Destiny: A legendary problem and the battle over who solved it". The New...