Categorial grammar

In mathematics, a Poisson manifold is a smooth manifold $M$ equipped with a bilinear map $\{\cdot ,\cdot {\}}_M$ (called a Poisson bracket) on the algebra $C^{\mathrm{\infty }}(M)$ of smooth functions on $M$ such that $(C^{\mathrm{\infty }}(M),\{\cdot ,\cdot {\}}_M)$ is a Poisson algebra. One usually denotes a Poisson manifold by the ordered pair $(M,\{\cdot ,\cdot {\}}_M)$. Since their introduction by André Lichnerowicz in 1977, the subjects of Poisson geometry and the cohomology of Poisson manifolds have developed into a wide field of research, which includes modern-day non-commutative geometry.

It is a fact that every symplectic manifold is a Poisson manifold but not vice-versa. This will be explained in Section 2.

Definition

A Poisson bracket (or Poisson structure) on a smooth manifold $M$ is a bilinear map

$\{\cdot ,\cdot \}:C^{\mathrm{\infty }}(M)\times C^{\mathrm{\infty }}(M)\to C^{\mathrm{\infty }}(M)$

that satisfies the following three properties:

• It is skew-symmetric: $\{f,g\}=-\{g,f\}$.
• It obeys the Jacobi Identity: $\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0$.
• It obeys Leibniz's Rule with respect to the first argument: $\{fg,h\}=f\{g,h\}+g\{f,h\}$.

By skew-symmetry, the Poisson bracket automatically satisfies Leibniz's Rule with respect to the second argument. The last property basically states that the map $f\mapsto \{f,g\}$ is a derivation on $C^{\mathrm{\infty }}(M)$ for any fixed $g\in C^{\mathrm{\infty }}(M)$. Every derivation $\delta$ on $C^{\mathrm{\infty }}(M)$ can be written as a directional derivative $[\delta (f)](x)=(\mathrm{d}f{)}_x((X_{\delta }{)}_x)$, where $x\in M$, for some vector field $X_{\delta }$. It follows that for $g\in C^{\mathrm{\infty }}(M)$, we obtain a vector field $X_g$ such that $\{f,g\}(x)=(\mathrm{d}f{)}_x((X_g{)}_x)$, where $x\in M$ (written more briefly, $\{f,g\}=\mathrm{d}f(X_g)$). The vector field $X_g$ is called the Hamiltonian vector field corresponding to $g$. Notice that

$⟨\mathrm{d}f,X_g⟩=X_g(f)=\{f,g\}=-X_f(g)=-⟨\mathrm{d}g,X_f⟩$,

where $⟨\cdot ,\cdot ⟩$ is the pairing between the cotangent and tangent bundles of $M$. Therefore, $\{f,g\}$ depends only on the differentials $\mathrm{d}f$ and $\mathrm{d}g$. Any Poisson bracket yields a map from the cotangent bundle to the tangent bundle that sends $\mathrm{d}f$ to $X_f$.

The Poisson Bivector

Given a Poisson manifold $(M,\{\cdot ,\cdot {\}}_M)$, the pairing between the cotangent and tangent bundles yields a bivector field $\eta$ on $M$, called the Poisson bivector field. The Poisson bivector field is a contravariant skew-symmetric 2-tensor field $\eta \in \mathrm{\Gamma }\left({\bigwedge }^{2}TM\right)$ that satisfies the following:

$\mathrm{\forall }x\in M:\{f,g{\}}_M(x)=⟨(\mathrm{d}f{)}_x\otimes (\mathrm{d}g{)}_x,{\eta }_x⟩$.

Conversely, given a smooth bivector field $\eta$ on $M$, we can use the formula above to define a skew-symmetric bracket $\{\cdot ,\cdot {\}}_{\eta }$ that obeys Leibniz's rule with respect to each argument. However, we cannot claim that $\{\cdot ,\cdot {\}}_{\eta }$ is a Poisson bracket because the Jacobi Identity may not hold (in this case, we call $\{\cdot ,\cdot {\}}_{\eta }$ an almost-Poisson structure). Indeed, $\{\cdot ,\cdot {\}}_{\eta }$ is a Poisson bracket if and only if the Schouten–Nijenhuis bracket $[\eta ,\eta ]$ equals zero.

In terms of local coordinates, the bivector field at a point $x=(x_1,\dots ,x_m)$ can be expressed as

$\mathrm{\forall }x\in M:{\eta }_x=\sum _{i,j=1}^m{\eta }_{i,j}(x)\cdot {\left(\frac{\partial }{\partial x_i}\right)}_x\otimes {\left(\frac{\partial }{\partial x_j}\right)}_x$,

so that

$\{f,g\}(x)=⟨(\mathrm{d}f{)}_x\otimes (\mathrm{d}g{)}_x,{\eta }_x⟩=\sum _{i,j=1}^m{\eta }_{i,j}(x){\left(\frac{\partial f}{\partial x_i}\right)}_x{\left(\frac{\partial g}{\partial x_j}\right)}_x$.

For a symplectic manifold $(M,\omega )$, we can define a bivector field $\eta$ on $M$ using the pairing between the cotangent and tangent bundles given by the symplectic form $\omega$. This pairing is well-defined because $\omega$ is nondegenerate. Hence, the difference between a symplectic manifold and a Poisson manifold is that the symplectic form is regular (of full rank) everywhere but the Poisson bivector field need not have full rank everywhere. When the Poisson bivector field is zero everywhere, we call it the trivial Poisson structure.

Poisson Maps

A Poisson map from a Poisson manifold $(M,\{\cdot ,\cdot {\}}_M)$ to another Poisson manifold $(N,\{\cdot ,\cdot {\}}_N)$ is defined to be a smooth map $\phi :M\to N$ that respects the Poisson structures in the following sense:

$\{f_1,f_2{\}}_N\circ \phi =\{f_1\circ \phi ,f_2\circ \phi {\}}_M$.

A Poisson map may be viewed as a morphism in the category of Poisson manifolds.

The Product of Poisson Manifolds

Given two Poisson manifolds $(M,\{\cdot ,\cdot {\}}_M)$ and $(N,\{\cdot ,\cdot {\}}_N)$, a Poisson bracket may be defined on the product manifold $M\times N$. Letting $f_1$ and $f_2$ be two smooth functions defined on $M\times N$, one can define a new Poisson bracket $\{\cdot ,\cdot {\}}_{M\times N}$ in terms of $\{\cdot ,\cdot {\}}_M$ and $\{\cdot ,\cdot {\}}_N$ as follows:

$\{f_1,f_2{\}}_{M\times N}(x,y)=\{f_1(x,\cdot ),f_2(x,\cdot ){\}}_N(y)+\{f_1(\cdot ,y),f_2(\cdot ,y){\}}_M(x)$,

where $x\in M$ and $y\in N$ are to be held constant. In other words, if

$f(\cdot ,\cdot ):M\times N\to \mathbf{R}$,

then both

$f(x,\cdot ):N\to \mathbf{R}$

and

$f(\cdot ,y):M\to \mathbf{R}$

are implied.

The Symplectic Leaves of a Poisson Structure

A Poisson manifold $(M,\{\cdot ,\cdot {\}}_M)$ can be split into a collection of symplectic leaves. This splitting arises from the foliation of disjoint regions of $M$ where the Poisson bivector field has constant rank. Each leaf of the foliation is thus an even-dimensional sub-manifold of $M$ that is itself a symplectic manifold. Distinct symplectic leaves may have different dimensions. Two points lie in the same leaf if and only if they are joined by a piecewise-smooth curve where each piece is the integral curve of a Hamiltonian vector field. The relation "piecewise-connected by integral curves of Hamiltonian fields" is an equivalence relation on $M$, and the equivalence classes of this equivalence relation are the symplectic leaves.

Example (Lie-Poisson Manifold)

If $\mathfrak{g}$ is a finite-dimensional Lie algebra and ${\mathfrak{g}}^{*}$ is its dual vector space, then the Lie bracket induces a Poisson structure on ${\mathfrak{g}}^{*}$.

More precisely, we identify the cotangent bundle of the manifold ${\mathfrak{g}}^{*}$, i.e., the dual of ${\mathfrak{g}}^{*}$ with the given Lie algebra $\mathfrak{g}$. Then for two functions $f_1$ and $f_2$ on ${\mathfrak{g}}^{*}$, and a point $\mathbf{x}\in {\mathfrak{g}}^{*}$, we may define

$\{f_1,f_2\}(\mathbf{x}):=⟨[(\mathrm{d}{f}_1{)}_{\mathbf{x}},(\mathrm{d}{f}_2{)}_{\mathbf{x}}],\mathbf{x}⟩$,

where the Lie bracket $[\cdot ,\cdot ]$ is computed in $\mathfrak{g}$ through the isomorphism:

$\mathrm{d}f\in {\mathfrak{g}}^{**}\cong \mathfrak{g}$.

If ${\mathbf{e}}_k$ are local coordinates on $\mathfrak{g}$, then the Poisson bivector field is given by

${\eta }_{i,j}(\mathbf{x})=\sum _k{c}_{ij}^k⟨\mathbf{x},{\mathbf{e}}_k⟩$,

where the $c_{ij}^k$ are the structure constants of $\mathfrak{g}$.

The symplectic leaves of this Lie-Poisson manifold are the co-adjoint orbits of the Lie algebra used for the orbit method.

Complex Structure

A complex Poisson manifold is a Poisson manifold with a complex or almost complex structure $J$ such that the complex structure preserves the bivector:

$(J\otimes J)(\eta )=\eta$.

The symplectic leaves of a complex Poisson manifold are pseudo-Kähler manifolds.

See also

• Nambu-Poisson manifold
• Poisson-Lie group
• Poisson supermanifold

Notes

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References

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  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Errata and addenda J. Diff. Geom. 22 (1985), 255.