points-to-pieces transform

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#### Cohesive $\infty$-Toposes
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#### Discrete and concrete objects
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## Idea

In a [[cohesive (∞,1)-topos]] $\mathbf{H}$, the canonical [[natural transformation]]

$$
  \flat \to \Pi
$$

from the [[flat modality]] to the [[shape modality]] may be thought of as sending "points to the pieces in which they sit".

## Definition and basic properties

Notice the existence of the following canonical [[natural transformations]] induced from the structure of a cohesive topos (a special case of the construction at _[[unity of opposites]]_).

+-- {: .num_defn #TransformationFromPointsToPieces}
###### Definition

Given a [[cohesive topos]] $\mathcal{E}$ with ($ʃ \dashv \flat$) its ([[shape modality]] $\dashv$ [[flat modality]])-[[adjunction]], then the [[natural transformation]]

$$
  \flat X \longrightarrow X \longrightarrow ʃ X
$$

(given by the [[composition]] of the $\flat$-[[counit of a comonad|counit]] followed by the $ʃ$-[[unit of a monad|unit]]) may be called the transformation **from points to their pieces** or the **points-to-pieces-transformation**, for short.

If this is an [[epimorphism]] for all $X$, we say that _pieces have points_ or that the _[[Nullstellensatz]]_ is verified.

=--

+-- {: .num_remark }
###### Remark

The $(f^\ast \dashv f_\ast)$-[[adjunct]] of the transformation from pieces to points, def. \ref{TransformationFromPointsToPieces},

$$
  \flat X \longrightarrow X \longrightarrow ʃ X
$$

is (by the rule of forming right [[adjuncts]] by first applying the [[right adjoint]] functor and then precomposing with the [[unit of an adjunction|unit]] and by the fact that the [[adjunct]] of a [[unit of an adjunction|unit]] is the [[identity]]) the map

$$
  (f_\ast X \longrightarrow f_! X)
  \coloneqq
  \left(
  f_\ast X \longrightarrow f_\ast f^\ast f_! X \stackrel{\simeq}{\longrightarrow} f_!X
  \right)
  \,.
$$

Observe that going backwards by applying $f^\ast$ to this and postcomposing with the $(f^\ast \dashv f_\ast)$-[[counit of an adjunction|counit]] is equivalent to just applying $f^\ast$, since by [[idempotent monad|idempotency]] of $\flat$ the counit is an [[isomorphism]] on the [[discrete object]] $f^\ast f_! X$. Therefore the points-to-pieces transformation and its adjunct are related by

$$
  \left(
    \flat X \longrightarrow X \longrightarrow ʃ X
  \right)
  =
  f^\ast \left(
    f_\ast X \longrightarrow f_! X
  \right).
$$

Observe then finally that since $f^\ast$ is a [[full and faithful functor|full and faithful]] [[left adjoint|left]] and [[right adjoint]], the points-to-pieces transform is an [[epimorphism]]/[[isomorphism]]/[[monomorphism]] precisely if its [[adjunct]] $f_\ast X \longrightarrow f_! X$ is, respectively.

If this adjunct

$$
  \flat X
    \overset{\epsilon^{\flat}}{\longrightarrow}
  X
    \overset{\eta^\sharp}{\longrightarrow}
  \sharp X
$$

is a monomorphism, we say that _[[discrete objects]] are [[concrete objects|concrete]]_.

=--

## Relation to points to co-pieces

+-- {: .num_prop #PiecesHavePointsIfDiscreteObjectsAreConcrete}
###### Proposition
**([[pieces have points]] iff [[discrete objects are concrete]])**

For a [[cohesive topos]] $\mathbf{H}$, the the following two conditions are equivalent:

1. _pieces have points_, i.e. $\flat X \to X \to ʃ X$ is an [[epimorphism]] for all $X \in \mathbf{H}$;

1. _[[discrete objects]] are [[concrete objects|concrete]]_, i.e. $\flat X \overset{ \eta^{\sharp}_{\flat X} }{\longrightarrow} \sharp \flat X$ is a [[monomorphism]].

=--

See at _[[cohesive topos]]_ [this prop.](cohesive+topos#PiecesHavePointsEquivalentToDiscreteObjectsAreConcrete).

## Relation to Aufhebung of the initial opposition
 {#RelationToAufhebung}

For a cohesive 1-topos, if the pieces-to-points transform is an [[epimorphism]] then there is [[Aufhebung]] of the initial opposition $(\emptyset \dashv \ast)$ in that $\sharp \emptyset \simeq \emptyset$ ([Lawvere-Menni 15, lemma 4.1](#LawvereMenni15), see also [Shulman 15, section 3](#Shulman15)). Conversely, if the [[base topos]] is a [[Boolean topos]], then this Aufhebung implies that the pieces-to-points transform is an [[epimorphism]] ([Lawvere-Menni 15, lemma 4.2](#LawvereMenni15)).

## Examples

### Bundle equivalence and concordance
 {#BundleEquivalenceAndConcordance}

Given an [[∞-group]] $G$ in a [[cohesive (∞,1)-topos]] $\mathbf{H}$, with [[delooping]] $\mathbf{B}G$, then for any other object $X$ the [[∞-groupoid]] $\mathbf{H}(X,\mathbf{B}G)$ is that of $G$-[[principal ∞-bundles]] with [[equivalences]] between them. Alternatively one may form the [[internal hom]] $[X,\mathbf{B}G]$. Applying the [[shape modality]] to this yields the $\infty$-groupoid $\mathbf{H}^\infty(X,\mathbf{B}G) \coloneqq  ʃ [X,\mathbf{B}G]$ of $G$-principal $\infty$-bundles and [[concordances]] between them. Alternatively, the [[flat modality]] applied to the internal hom is again just the external hom $\flat [X,\mathbf{B}G] \simeq \mathbf{H}(X,\mathbf{B}G)$.

In conclusion, in this situation the points-to-pieces transform is the canonical map

$$
  \mathbf{H}(X,\mathbf{B}G) \longrightarrow \mathbf{H}^\infty(X,\mathbf{B}G)
$$

from $G$-principal $\infty$-bundles with bundle equivalences between them, to $G$-principal $\infty$-bundles with concordances between them.

### In global equivariant homotopy theory
  {#InGlobalEquivariantHomotopyTheory}

In [[global equivariant homotopy theory]]  an incarnation of the points to pieces transform is the comparison map from [[homotopy quotients]] to ordinary [[quotients]]

$$
  X//G \longrightarrow X/G
$$

which in terms of the [[Borel construction]] is induced by the map $E G \to \ast$

$$
  E G \times_G X \longrightarrow \ast \times_G X = X/G
  \,.
$$

(see at [[global equivariant homotopy theory]] [this prop.](global+equivariant+homotopy+theory#OrdinaryQuotientAndHomotopyQuotientViaCohesion))

### In tangent cohesion: the differential cohomology diagram

In a [[tangent cohesive (∞,1)-topos]] on [[stable homotopy types]] the points-to-pieces transform is one stage in a natural hexagonal [[long exact sequence]], the _[[differential cohomology diagram]]_. See there for more.

### Comparison map between algebraic and topological K-theory

Applied to [[stable homotopy types]] in $Stab(\mathbf{H}) \hookrightarrow T\mathbf{H}$ the [[tangent cohesive (∞,1)-topos]] which arise from a [[symmetric monoidal (∞,1)-category]] $V \in CMon_\infty(Cat_\infty(\mathbf{H}))$ [[internal (∞,1)-category|internal]] to $\mathbf{H}$ under internal [[algebraic K-theory of a symmetric monoidal (∞,1)-category]], the points-to-pieces transform interprets as the [[comparison map between algebraic and topological K-theory]]. See there for more

### In infinitesimal cohesion

In [[infinitesimal cohesion]] the points-to-pieces transform is an [[equivalence]].

## Related concepts

[[!include cohesion - table]]

## References

* {#LawvereMenni15} [[William Lawvere]], [[Matías Menni]], _Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness_, Theory and Applications of Categories, Vol. 30, 2015, No. 26, pp 909-932. ([TAC](http://www.tac.mta.ca/tac/volumes/30/26/30-26abs.html))

* {#Shulman15} [[Mike Shulman]], _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))
[[!redirects points-to-pieces transforms]]

[[!redirects points-to-pieces transformation]]
[[!redirects points-to-pieces transformations]]

[[!redirects points-to-pieces-transformation]]
[[!redirects points-to-pieces-transformations]]

[[!redirects points-to-pieces-transform]]
[[!redirects points-to-pieces-transforms]]

[[!redirects pieces have points]]
[[!redirects pieces-have-points]]

[[!redirects discrete objects are concrete]]


pointwise continuous function

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## Idea ##

The concept of [[continuous functions]] familiar from [[epsilontic analysis]].

## Definition ##

+-- {: .num_defn #InRealNumbers}
###### Definition
**(pointwise continuous function in the real numbers)

Let $\mathbb{R}$ be the [[real numbers]] and let

$$\mathbb{R}_{+} \coloneqq \{a \in \mathbb{R} \vert 0 \lt a\}$$

be the positive elements in $\mathbb{R}$. A function $f:\mathbb{R} \to \mathbb{R}$ is __continuous at a point__ $c \in \mathbb{R}$ if

$$isContinuousAt(f, c) \coloneqq \forall \epsilon \in \mathbb{R}_{+}. \exists \delta \in \mathbb{R}_{+}. \forall x \in \mathbb{R}. (\vert x - c \vert \lt \delta) \to (\vert f(x) - f(c) \vert \lt \epsilon)$$

$f$ is __pointwise continuous__ in $\mathbb{R}$ if it is continuous at all points $c$:
$$isPointwiseContinuous(f) \coloneqq \forall c \in \mathbb{R}. isContinuousAt(f, c)$$

=--

+-- {: .num_defn #BetweenArchimedeanOrderedFields}
###### Definition
**(pointwise continuous function between Archimedean ordered fields)**

Let $F$ and $K$ be [[Archimedean ordered fields]] and let

$$F_{+} \coloneqq \{a \in F \vert 0 \lt a\} \quad K_{+} \coloneqq \{a \in K \vert 0 \lt a\}$$

be the positive elements in $F$ and $K$. A function $f:F \to K$ is __continuous at a point__ $c \in F$ if

$$isContinuousAt(f, c) \coloneqq \forall \epsilon \in K_{+}. \forall x \in F. \exists \delta \in F_{+}. (\max(x - c, c - x) \lt \delta) \to (\max(f(x) - f(c), f(c) - f(x)) \lt \epsilon)$$

$f$ is __pointwise continuous__ if it is continuous at all points $c$:
$$isPointwiseContinuous(f) \coloneqq \forall c \in F. isContinuousAt(f, c)$$

=--

+-- {: .num_defn #BetweenMetricSpaces}
###### Definition
**(pointwise continuous function between metric spaces)**

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces]], and let

$$\mathbb{R}_{+} \coloneqq \{a \in \mathbb{R} \vert 0 \lt a\}$$

be the positive real numbers. A function $f:X \to Y$ is __continuous at a point__ $c \in X$ if

$$isContinuousAt(f, c) \coloneqq \forall \epsilon \in \mathbb{R}_{+}. \forall x \in X. \exists \delta \in \mathbb{R}_{+}. d_X(x, c) \lt \delta \to d_Y(f(x), f(c)) \lt \epsilon)$$

$f$ is __pointwise continuous__ if it is continuous at all points $c$:
$$isPointwiseContinuous(f) \coloneqq \forall c \in X. isContinuousAt(f, c)$$

=--

+-- {: .num_defn #BetweenUniformSpaces}
###### Definition
**(pointwise continuous function between uniform spaces)**

Let $(X, \mathcal{U}(X), \approx)$ and $(Y, \mathcal{U}(Y), \approx)$ be [[uniform spaces]]. A function $f:X \to Y$ is __continuous at a point__ $c \in X$ if

$$isContinuousAt(f, c) \coloneqq \forall E \in \mathcal{U}(Y). \forall x \in X. \exists D \in \mathcal{U}(X). x \approx_D c \to f(x) \approx_E f(c))$$

$f$ is __pointwise continuous__ if it is continuous at all points $c$:
$$isPointwiseContinuous(f) \coloneqq \forall c \in X. isContinuousAt(f, c)$$

=--

+-- {: .num_defn #BetweenPreconvergenceSpaces}
###### Definition
**(pointwise continuous function between preconvergence spaces)**

Let $(X, \mathcal{F}(X), \mathrm{isLimit}_X)$ and $(Y, \mathcal{F}(Y), \mathrm{isLimit}_Y)$ be [[preconvergence spaces]]. A function $f:X \to Y$ is __continuous at a point__ $c \in X$ if

$$isContinuousAt(f, c) \coloneqq \forall F \in \mathcal{F}(X).\mathrm{isLimit}_X(F, c) \to \mathrm{isLimit}_Y(f(F), f(c)))$$

$f$ is __pointwise continuous__ if it is continuous at all points $c$:
$$isPointwiseContinuous(f) \coloneqq \forall c \in X. isContinuousAt(f, c)$$
=--

+-- {: .num_defn #InFunctionLimitSpaces}
###### Definition
**(pointwise continuous function in function limit spaces)**

Let $T$ be a [[Hausdorff function limit space]], and let $S \subseteq T$ be a [[subset]] of $T$. A function $f:S \to T$ is __continuous at a point__ $c \in S$ if the [[limit of a function|limit of $f$ approaching $c$]] is equal to $f(c)$.

$$\lim_{x \to c} f(x) = f(c)$$

$f$ is __pointwise continuous__ on $S$ if it is continuous at all points $c \in S$:

$$isPointwiseContinuous(f) \coloneqq \forall c \in S. \lim_{x \to c} f(x) = f(c)$$
=--

### As structure

In [[dependent type theory]], one could change the [[universal quantifiers]] and [[existential quantifiers]] in the definition of uniformly continuous function into [[dependent product types]] and [[dependent sum types]].

\begin{definition}
Let $\mathrm{R}_+ \coloneqq \sum_{x:\mathbb{R}} \epsilon \gt 0$ denote the positive real numbers. Given [[metric spaces]] $(X, d_X)$ and $(Y, d_Y)$, a **pointwise continuous function** between $X$ and $Y$ is a function $f:X \to Y$ between their underlying sets with a dependent function which says:

> For all elements $a:X$ and for all positive real number $\epsilon \gt 0$, there is as structure a positive real number $\delta \gt 0$ such that for all elements $b:X$, $\delta_Y(f(a), f(b))$ is less than $\epsilon$ whenever $\delta_X(a, b)$ is less than $\delta$

$$\prod_{a:X} \prod_{\epsilon:\mathrm{R}_+} \sum_{\delta:\mathbb{R}_+} \prod_{b:X} (\delta_X(a, b) \lt \delta) \to (\delta_Y(f(a), f(b)) \lt \epsilon)$$

By the [[type theoretic axiom of choice]], which is simply the distributivity of [[dependent function types]] over [[dependent sum types]], this is the same as saying that

> There exists as structure a function $\omega:X \to (\mathrm{R}_+ \to \mathrm{R}_+)$ such that for all elements $a:X$, for all positive real numbers $\epsilon \gt 0$ and for all elements $b:X$, $\delta_Y(f(a), f(b))$ is less than $\epsilon$ whenever $\delta_X(a, b)$ is less than $\omega(\epsilon)$

$$\sum_{\omega:X \to (\mathrm{R}_+ \to \mathrm{R}_+)} \prod_{a:X} \prod_{\epsilon:\mathrm{R}_+} \prod_{b:X} (\delta_X(a, b) \lt \omega(\epsilon)) \to (\delta_Y(f(a), f(b)) \lt \epsilon)$$
\end{definition}

There exists a similar definition for [[uniform spaces]]:

\begin{definition}
Given uniform spaces $(X, \mathcal{U}(X), \approx)$ and $(Y, \mathcal{U}(Y), \approx)$, a **pointwise continuous function** between $X$ and $Y$ is a function $f:X \to Y$ with a dependent function which says:

> For all elements $x:A$ and for all [[entourages]] $E:\mathcal{U}(Y)$, there is as structure an entourage $D:\mathcal{U}(X)$ such that for all elements $b:X$, $f(a) \approx_{E} f(b)$ whenever $a \approx_{D} b$

$$\prod_{a:X} \prod_{E:\mathcal{U}(Y)} \sum_{D:\mathcal{U}(X)} \prod_{b:X} (a \approx_{D} b) \to (f(a) \approx_{E} f(b))$$

By the [[type theoretic axiom of choice]], which is simply the distributivity of [[dependent function types]] over [[dependent sum types]], this is the same as saying that

> There exists as structure a function $\omega:X \to (\mathcal{U}(Y) \to \mathcal{U}(X))$ between the sets of entourages such that for all elements $a:X$ and entourages $E:\mathcal{U}(Y)$ and for all elements $b:X$, $f(a) \approx_{E} f(b)$ whenever $a \approx_{\omega(E)} b$

$$\sum_{\omega:X \to (\mathcal{U}(Y) \to \mathcal{U}(X))} \prod_{a:X} \prod_{E:\mathcal{U}(Y)} \prod_{b:X} (a \approx_{\omega(a, E)} b) \to (f(a) \approx_{E} f(b))$$
\end{definition}

## See also ##

* [[limit of a function]]
* [[differentiable function]]
* [[uniformly continuous function]]

[[!redirects pointwise continuous]]
[[!redirects pointwise continuous functions]]


pointwise convergence

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#Contents#
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## Idea

(...)

## Related concepts

* [[uniform convergence]]

## References

See also:

* Wikipedia, _[Pointwise convergence](https://en.wikipedia.org/wiki/Pointwise_convergence)_

[[!redirects topology of pointwise convergence]]


pointwise order

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# Contents
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## Idea

The pointwise order is the canonical [[order]] on the set of functions into a [[poset]] or [[preorder]].

Using [[negative thinking]], this can be seen as the [[(0,1)-category|(0,1)-decategorification]] of the concept of [[category of functors]].
Just as well, it can be seen as the [[enriched functor category]] for categories enriched in [[truth values]].

## Definition

+-- {: .num_defn}
###### Definition

Let $X$ be a set, and let $Y$ be a [[preorder|preordered set]]. Given $f,g:X\to Y$, we say that $f\le g$ in the **pointwise order** or **product order** if and only of for every $x\in X$ we have $f(x)\le g(x)$.

=--

The terminology "product order" comes from the fact that the order defined above can be seen as the one of the object
$$
\prod_{x\in X} Y
$$
in the category of preorders (which has products). The underlying set of the object above is indeed naturally isomorphic to the set of functions $X\to Y$.

## 2-cells

Just as [[Cat]] is naturally a [[2-category]], with 2-cells given by [[natural transformations]] (i.e. [[morphisms]] of [[functors]]), many categories of preorders (such as [[Pos]]) are naturally [[locally posetal 2-categories]], with the 2-cells given by the pointwise order. That is, given $f,g:X\to Y$, we draw a 2-cell $f\Rightarrow g$ if and only if $f\le g$ in the pointwise order.
If the morphisms are chosen to be [[monotone]], this choice of 2-cells gives indeed the structure of a [[2-category]].

## See also

* [[Pos]]
* [[locally posetal 2-category]]
* [[(0,1)-category]]

[[!redirects product order]]
[[!redirects product ordering]]
[[!redirects pointwise ordering]]
[[!redirects pointwise ordered]]
[[!redirects ordered pointwise]]


Poiss

The [[category]] of [[Poisson algebra]]s.

category: category


Poisson algebra

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## Definition

+-- {: .num_defn}
###### Definition

A **Poisson algebra** is

* a [[module]] $A$ over some [[field]] or other [[commutative ring]] $k$,

* equipped with the structure ${\cdot}\colon A \otimes_k A \to A$ of a commutative [[associative algebra]];

* and equipped with the bracket $[-,-]\colon A \otimes_k A \to A$ of a [[Lie algebra]];

* such that for every $a \in A$ we have that $[a,-]\colon A \to A$ is a [[derivation]] of $(A,\cdot)$.

=--

The definition makes sense, but is not standardly used, also for the more general case when the product $\cdot$ is not necessarily commutative (it is often however taken in the sense of commutative internally to a symmetric monoidal category, say of chain complexes, graded vector spaces or supervector spaces).

Poisson algebras form a category [[Poiss]].

+-- {: .num_defn}
###### Definition

The [[opposite category]] of that of (commutative) real Poisson algebras can be identified with the category of **[[classical mechanical systems]]**

$$
  ClassMechSys := CPoiss^{op}
  \,.
$$

See there for more details.

=--

+-- {: .num_example}
###### Example

For $(X, \{-,-\})$ a [[Poisson manifold]] or $(X, \omega)$ a [[symplectic manifold]], the algebra of [[smooth function]]s $C^\infty(X, \mathbb{R})$ is naturally a Poisson algebra, thus may be regarded as an object in $ClassMechSys$. For classical mechanical systems of this form, we say that the [[manifold]] $X$ is the [[phase space]] of the system.

Generally, therefore, for $(A, \cdot,[-,-])$ a Poisson algebra, we may regard it as a formal dual to some generalized Phase space.

=--

+-- {: .num_remark}
###### Remark

For $(A, \cdot, \{-,-\})$ a Poisson algebra, $A$ together with its [[module]] $\Omega^1(A)$ of [[Kähler differential]]s naturally form a [[Lie-Rinehart pair]], with bracket given by

$$
  [d a, d b ] := d \{a,b\}
  \,.
$$

If the Poisson algebra comes from a [[Poisson manifold]] $X$, then this Lie-Rinehart pair is the [[Chevalley-Eilenberg algebra]] of the given [[Poisson Lie algebroid]] over $X$. We can therefore identify classical mechanical systems over a phase space manifold also with Poisson Lie algebroids.

=--

## Examples

### For a symplectic manifold
 {#FromSymplecticManifold}

+-- {: .num_defn}
###### Definition

A [[symplectic manifold]] $(X, \omega)$
canonically is a [[Poisson manifold]] $(X; \{-,-\})$ by
defining the Poisson bracket as follows.

By the symplectic structure,
to every [[smooth function]] $f \in C^\infty(X)$
is associated the correspinding [[Hamiltonian vector field]]
$v_f \in \Gamma(T X)$, defined, uniquely, by the equation

$$
  d_{dR} f = \iota_{v_f} \omega
  \,.
$$

In terms of this, the Poisson bracket is given by

$$
  \{f,g\} := \iota_{v_g} \iota_{v_f} \omega
  \,.
$$

=--

## Properties

### For a symplectic manifold

Let $(X, \omega)$ be a [[symplectic manifold]],
and $(X, \{-,-\})$ the corresponding [[Poisson manifold]]
as [above](FromSymplecticManifold).

Write
$\mathcal{P} := (C^\infty(X), \{-,-\})$ for the [[Lie algebra]]
underlying the Poisson algebra.

+-- {: .num_prop}
###### Proposition

This fits into a [[central extension]] of Lie algebras

$$
  \mathbb{R} \to \mathcal{P} \to Ham(X)
  \,,
$$

where $Ham(X) \subset \Gamma(T X)$ is the sub-Lie algebra of [[vector fields]] on the [[Hamiltonian vector fields]].

=--

+-- {: .proof}
###### Proof

Observe that the [[Hamiltonian]] function associated to a
[[Hamiltonian vector field]] is well-defined only up to addition of
a constant function.

=--

This is also called the **Kostant-Souriau central extension**
(see [Kostant 1970](#Kostant)).

## Related concepts

* A _[[polynomial Poisson algebra]]_ is one whose underlying commutative algebra is a [[polynomial algebra]].

*  A __[[Poisson manifold]]__ is a [[smooth manifold]] whose associatve algebra of [[smooth function]]s with values in the [[real line]] $\mathbb{R}$ that is equipped with the structure of a Poisson algebra (over $\mathbb{R}$).

* A [[Poisson n-algebra]] is a Poisson algebra in chain complexes with a bracket of degree $1-n$.

  *  A [[Gerstenhaber algebra]] is a Poisson 2-algebra.

* A [[Coisson algebra]] is essentially a Poisson algebra [[internalization|internal to]] [[D-module]]s.

* A [[Jordan–Lie algebra]] is a non-associative (quantum) Poisson algebra.

* The [[quantomorphism group]] is the [[Lie group]] that integrates the Poisson [[Lie bracket]]. Over a [[symplectic vector space]] this contains notably the corresponding [[Heisenberg group]].

[[!include Isbell duality - table]]

## References

* [[Yvette Kosmann-Schwarzbach]],  _Poisson algebra_, article in _Encyclopedia of mathematics_, ([pdf](http://www.math.polytechnique.fr/~kosmann/palg.pdf))

* [[Klaas Landsman]], _[[Mathematical Topics Between Classical and Quantum Mechanics]]_

* N. Chriss, [[Victor Ginzburg]], _Complex geometry and representation theory_

* [[Peter Olver]], _Equivalence, invariants, and symmetry_, Cambridge Univ. Press 1995

* {#Kostant} [[Bertram Kostant]], _Quantization and unitary representations. I. Prequantization_, In _Lectures in Modern Analysis and Applications, III_, pages 87–208. Lecture Notes in Math., Vol. 170. Springer, Berlin (1970)

Discussion of [[Lie integration]] of the Poisson brackets on non-compact manifolds, by restriction to [[compactly supported functions]]:

* Augustin Banyaga, Paul Donato, *Some remarks on the integration of the Poisson algebra*, Journal of Geometry and Physics **19** 4 (1996) 368-378 [doi:10.1016/0393-0440(95)00039-9]

[[!redirects Poisson algebra]]
[[!redirects Poisson algebras]]


Poisson bracket Lie n-algebra

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## Idea

The [[Lie n-algebra]] that generalizes the [[Poisson bracket]] from [[symplectic geometry]] to [[n-plectic geometry]]: the _Poisson bracket $L_\infty$-algebra of local observables_ in [[higher prequantum geometry]].

More discussion is [here](n-plectic+geometry#PoissonLInfinityAlgebras) at _[[n-plectic geometry]]_.

Applied to the symplectic current (in the sense of [[covariant phase space]] theory, [[de Donder-Weyl field theory]]) this is the higher [[current algebra]] (see there) of [[conserved currents]] of a [[prequantum field theory]].

## Definition
 {#Definition}

Throughout, Let $X$ be a [[smooth manifold]], let $n \geq 1$ a natural number  and $\omega \in \Omega^{n+1}_{cl}(X)$ a closed [[differential n-form|differential (n+1)-form]] on $X$. The pair $(X,\omega)$ we may regard as a [[pre-n-plectic manifold]].

We define two [[L-∞ algebras]] defined from this data and discuss their [[equivalence]]. Either of the two or any further one equivalent to the two is the _Poisson bracket Lie $n$-albebra_ of $(X,\omega)$. The first definition is due to ([Rogers 10](#Rogers10)), the second due to ([FRS 13b](#FRS13b)). Here in notation we follow ([FRS 13b](#FRS13b)).

+-- {: .num_defn #HamiltonianFormsAndVectorFields}
###### Definition

Write

$$
  Ham^{n-1}(X) \subset Vect(X) \oplus \Omega^{n-1}(X)
$$

for the subspace of the [[direct sum]] of [[vector fields]] $v$ on $X$ and [[differential n-form|differential (n-1)-forms]] $J$ on $X$ satisfying

$$
  \iota_v  \omega + \mathbf{d} J = 0
  \,.
$$

We call these the _pairs of [[Hamiltonian forms]] with their [[Hamiltonian vector fields]]_.

=--

([FRS 13b, def. 2.1.3](#FRS13b))

+-- {: .num_defn #PoissonBracketLienAlgebra}
###### Definition

The [[L-∞ algebra]] $L_\infty(X,\omega)$ has as underlying [[chain complex]] the truncated and modified [[de Rham complex]]

$$
  \Omega^0(X)
  \stackrel{\mathbf{d}}{\to}
  \Omega^1(X)
  \stackrel{\mathbf{d}}{\to}
  \cdots
  \stackrel{\mathbf{d}}{\to}
  \Omega^{n-2}(X)
  \stackrel{(0,\mathbf{d})}{\longrightarrow}
  Ham^{n-1}(X)
$$

with the Hamiltonian pairs, def. \ref{HamiltonianFormsAndVectorFields}, in degree 0 and with the 0-forms ([[smooth functions]]) in degree $n-1$, and its non-vanishing $L_\infty$-brackets are as follows:

* $l_1(J) = \mathbf{d}J$

* $l_{k \geq 2}(v_1 + J_1, \cdots, v_k + J_k) = - (-1)^{\left(k+1 \atop 2\right)} \iota_{v_1 \wedge \cdots \wedge v_k}\omega$.

=--

([FRS 13b, prop. 3.1.2](#FRS13b))

+-- {: .num_defn #PoissondgAlgebra}
###### Definition

Let $\overline{A}$ be any [[Cech cohomology|Cech]]-[[Deligne cohomology|Deligne]]-[[cocycle]] relative to an [[open cover]] $\mathcal{U}$ of $X$, which gives a [[prequantum n-bundle]] for $\omega$. The [[L-∞ algebra]] $dgLie_{Qu}(X,\overline{A})$ is the [[dg-Lie algebra]] (regarded as an $L_\infty$-algebra) whose underlying [[chain complex]] is

$dgLie_{Qu}(X,\overline{A})^0 = \{v+ \overline{\theta}  \in Vect(X)\oplus Tot^{n-1}(\mathcal{U}, \Omega^\bullet) \;\vert\; \mathcal{L}_v \overline{A} = \mathbf{d}_{Tot}\overline{\theta}\}$;

$dgLie_{Qu}(X,\overline{A})^{i \gt 0} = Tot^{n-1-i}(\mathcal{U},\Omega^\bullet)$

with [[differential]] given by $\mathbf{d}_{Tot}$ (where $Tot$ refers to [[total complex]] of the Cech-de Rham [[double complex]]).

The non-vanishing dg-Lie bracket on this complex are defined to be

* $[v_1 + \overline{\theta}_1, v_2 + \overline{\theta}_2] = [v_1, v_2] + \mathcal{L}_{v_1}\overline{\theta}_2 - \mathcal{L}_{v_2}\overline{\theta}_1$;

* $[v+ \overline{\theta}, \overline{\eta}] = - [\eta, v + \overline{\theta}] = \mathcal{L}_v \overline{\eta}$.

=--

([FRS 13b, def./prop. 4.2.1](#FRS13b))

+-- {: .num_prop #ComparisonTheorem}
###### Proposition

There is an [[equivalence]] in the [[model structure for L-∞ algebras|homotopy theory of L-∞ algebras]]

$$
  f
  \colon
  L_\infty(X,\omega)
  \stackrel{\simeq}{\longrightarrow}
  dgLie_{Qu}(X,\overline{A})
$$

between the $L_\infty$-algebras of def. \ref{PoissonBracketLienAlgebra} and def. \ref{PoissondgAlgebra} (in particular def. \ref{PoissondgAlgebra} does not depend on the choice of $\overline{A}$) whose underlying [[chain map]] satisfies

* $f(v + J) = v - J|_{\mathcal{U}} + \sum_{i = 0}^n (-1)^i \iota_v A^{n-i}$.

=--

([FRS 13b, theorem 4.2.2](#FRS13b))

## Properties

### The extension theorem

+-- {: .num_defn #ExtensionTheorem}
###### Proposition

Given a [[pre n-plectic manifold]] $(X,\omega_{n+1})$, then the Poisson bracket Lie $n$-algebra $\mathfrak{Pois}(X,\omega)$ from [above](#Definition) is an [[L-infinity algebra cohomology|extension]] of the [[Lie algebra]] of [[Hamiltonian vector fields]] $Vect_{Ham}(X)$, def. \ref{HamiltonianFormsAndVectorFields} by the [[cocycle]] [[infinity-groupoid]] $\mathbf{H}(X,\flat \mathbf{B}^{n-1} \mathbb{R})$ for [[ordinary cohomology]] with [[real number]] [[coefficients]] in that there is a [[homotopy fiber sequence]] in the [[homotopy theory of L-infinity algebras]] of the form

$$
  \array{
    \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R})
    &\longrightarrow&
    \mathfrak{Pois}(X,\omega)
    \\
    && \downarrow
    \\
    && Vect_{Ham}(X,\omega)
    &\stackrel{\omega[\bullet]}{\longrightarrow}&
    \mathbf{B} \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R})
  }
  \,,
$$

where the [[cocycle]] $\omega[\bullet]$, when realized on the model of def. \ref{PoissonBracketLienAlgebra}, is degreewise given by by contraction with $\omega$.

=--

This is [FRS13b, theorem 3.3.1](#FRS13b).

As a corollary this means that the [[0-truncation]] $\tau_0 \mathfrak{Pois}(X,\omega)$ is a [[Lie algebra extension]] by [[de Rham cohomology]], fitting into a [[short exact sequence]] of [[Lie algebras]]

$$
  0 \to H^{d-1}_{dR}(X) \longrightarrow \tau_0 \mathfrak{Pois}(X,\omega) \longrightarrow Vect_{Ham}(X) \to 0
 \,.
$$

+-- {: .num_remark}
###### Remark

These kinds of extensions are known traditionally form [[current algebras]].

=--

## Related concepts

* [[higher Poisson structure]]

  * [[Poisson n-algebra]]

  * [[Nambu bracket]]

* [[current algebra]]

* [[geometry of physics -- prequantum geometry]]

[[!include slice automorphism groups in higher prequantum geometry - table]]

[[!include geometric quantization extensions - table]]

## References

The Poisson bracket $L_\infty$-algebra $L_\infty(X,\omega)$ was introduced in

* {#Rogers10} [[Chris Rogers]], _$L_\infty$ algebras from multisymplectic geometry_,  Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50  ([arXiv:1005.2230](http://arxiv.org/abs/1005.2230), [journal](http://link.springer.com/article/10.1007%2Fs11005-011-0493-x)).

* {#Rogers11} [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))

Discussion in the broader context of [[higher differential geometry]] and [[higher prequantum geometry]] is in

* {#FRS13a} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory|Higher $U(1)$-gerbe connections in geometric prequantization]]_, Rev. Math. Phys., Vol. 28, Issue 06, 1650012 (2016) ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))

* {#FRS13b} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))

* Nestor Leon Delgado, _Lagrangian field theories: ind/pro-approach and L-infinity algebra of local observables_ ([arXiv:1805.10317](https://arxiv.org/abs/1805.10317))

See also

* {#RitterSaemann15} [[Patricia Ritter]], [[Christian Sämann]], _Automorphisms of Strong Homotopy Lie Algebras of Local Observables_ ([arXiv:1507.00972](http://arxiv.org/abs/1507.00972))

[[!redirects Poisson bracket Lie n-algebras]]

[[!redirects Poisson-bracket Lie n-algebra]]
[[!redirects Poisson-bracket Lie n-algebras]]

[[!redirects Poisson Lie n-algebra]]
[[!redirects Poisson Lie n-algebras]]

[[!redirects Poisson L-∞ algebra]]
[[!redirects Poisson L-∞ algebras]]

[[!redirects Poisson bracket L-∞ algebra]]
[[!redirects Poisson brakcet L-∞ algebras]]

[[!redirects Poisson L-infinity algebra]]
[[!redirects Poisson L-infinity algebras]]

[[!redirects Poisson-bracket Lie n-algebra of local observables]]
[[!redirects Poisson-bracket Lie n-algebras of local observables]]

[[!redirects L-∞ algebra of local observables]]
[[!redirects L-∞ algebras of local observables]]

[[!redirects Poisson bracket Lie (p+1)-algebra]]
[[!redirects Poisson bracket Lie (p+1)-algebras]]


Poisson cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $(X, \pi)$ a [[Poisson manifold]], its _Poisson cohomology_ ([Lichnerowicz](#Lichnerowicz)) is equivalently the [[Lie algebroid cohomology]] $H^\bullet(CE(\mathfrak{P}))$ of the corresponding [[Poisson Lie algebroid]] $\mathfrak{P}$, hence the [[cochain cohomology]] of its [[Chevalley-Eilenberg algebra]] $\mathfrak{P}$ ([Huebschmann](#Huebschmann)).

Similarly if one generalizes from the the Poisson algebra of functions on a smooth manifold is generalized to any [[Poisson algebra]], the corresponding _Poisson cohomology_ is that of the corresponding Poisson [[Lie-Rinehart pair]].

## References

The notion of Poisson cohomology was introduced in

* [[André Lichnerowicz]], Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), 253-300 [MR0501133](http://www.ams.org/mathscinet-getitem?mr=0501133) [euclid](http://projecteuclid.org/euclid.jdg/1214433987)
 {#Lichnerowicz}

A textbook reference on Poisson cohomology is section 3.6 (for $H^0$ and $H^1$) and sec. 18.4 (in terms of a cohomology for Lie algebroids) in

* A. Cannas da Silva, [[Alan Weinstein]], _Geometric models for noncommutative algebras_, Amer. Math. Soc. 1999 (Berkeley Mathematics Lecture Notes) [pdf](http://math.berkeley.edu/%7Ealanw/Models.pdf) [errata](http://math.berkeley.edu/%7Ealanw/Modelserrata.pdf)

* Izu Vaisman, _Remarks on the Lichnerowicz-Poisson cohomology_, Annales de l'institut Fourier 40.4 (1990): 951-963. <http://eudml.org/doc/74907>.

The observation that Poisson cohomology is just the [[Lie algebroid cohomology]] of the corresponding [[Poisson Lie algebroid]] is originally due to 1990 article

* [[Johannes Huebschmann]], _Poisson cohomology and quantization_ J. Reine Angew. Math. __408__ (1990), 57-113, [MR1058984](http://www.ams.org/mathscinet-getitem?mr=1058984) [doi](http://dx.doi.org/10.1515/crll.1990.408.57), reprinted at [arXiv:1303.3903](http://arxiv.org/abs/1303.3903)
 {#Huebschmann}

A different flavour is variational Poisson cohomology

* A. De Sole, [[Victor Kac]], _Essential variational Poisson cohomology_, Comm. Math. Phys. __313__ (2012), no. 3, 837-864 [arxiv/1106.5882](http://arxiv.org/abs/1106.5882); _The variational Poisson cohomology_, Jpn.
J. Math. 8, (2013), 1-145 [arxiv/1106.0082](http://arxiv.org/abs/1106.0082)


Poisson geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Poisson geometry_ is primarily the study of [[Poisson manifolds]] and the [[foliation]] by [[symplectic leaves]], but more generally it involves a host of induced and related structures that appear notably in the study of [[deformation quantization]] of Poisson manifolds, such as their [[Poisson Lie algebroids]], [[coisotropic submanifolds]], [[symplectic Lie groupoids]] etc.

## Related concepts

* [[Poisson manifold]]

* [[symplectic geometry]]

## References

Monographs:

* [[Marius Crainic]], [[Rui Loja Fernandes]], *Lectures on Poisson Geometry*, Graduate Studies in Mathematics **217**, Amer. Math. Soc. (2021) [[ISBN:978-1-4704-6430-1](https://bookstore.ams.org/view?ProductCode=GSM/217)]

For more see the references at *[[Poisson manifold]]*.


##  Poisson Geometry and Mathematical Physics##

***
2016 Jan. 4-8, Tianjin at [Chern Institute, Nankai University]( http://www.nim.nankai.edu.cn/nim_e/index.htm)

Registration: 2016 Jan. 3

***

####Organizers####
[Chengming Bai](http://www.nim.nankai.edu.cn/about_us/member/baicm.htm)  (Chern Institute, Nankai University)

 [[Zhuo Chen]] (Tsinghua Universtiy)

[Yunhe Sheng](http://math.jlu.edu.cn/?mod=teacher&act=view&id=184) (Jilin University)

***

#### Scientific Committee ####

[Zhangju Liu](http://www.math.pku.edu.cn/static/interface/people.htm?uid=liuzj) (Peking University)

[Jianghua Lu](http://hkumath.hku.hk/~jhlu/) (The University of Hong Kong)

[Ping Xu](https://www.math.psu.edu/ping/) (Penn State University)

[Weiping Zhang](http://www.nim.nankai.edu.cn/nim_e/members/weiping.htm) (Chern Institute, Nankai University)

***

####Invited Participants####

[[Anton Alekseev]] (University of Geneva)

Chengming Bai (Chern Institute, Nankai University)

Yanhui Bi (Nanchang Hangkong University)

Francesco Bonechi (INFN Sezione di Firenze)

Liqiang Cai (Jilin University)

Xiongwei Cai (University of Luxembourg)

Xiaojun Chen (Sichuan University)

Youming Chen (Sichuan University)

Zhuo Chen (Tshinghua University)

Nicola Ciccoli (University of Perugia)

Lu Ding (Chinese Academy of Science)

Farkhod Eshmatov (Sichuan University)

Samuel Evens (University of Notre Dame)

Huiju Fan (Peking University)

Janusz Grabowski (Polish Academy of Sciences)

Wei Hong (Wuhan University)

Xingbiao Hu (Institute of Computational Mathematics and Scientific/Engineering Computing)

[[Yvette Kosmann-Schwarzbach]] (Ecole polytechnique, Palaiseau)

Honglei Lang (Peking University)

[[Camille Laurent-Gengoux]] (The University of Poitiers)

Hui Li (Suzhou University)

Si Li (Tshinghua University)

Yanpeng Li (Peking University)

Runliang Lin (Tsinghua University)

Jiefeng Liu (Jilin University)

Zhangju Liu (Peking University)

Jianghua Lu  (The University of Hong Kong)

[[Kirill Mackenzie]] (University of Sheffield)

Kentaro Mikami (Akita University)

Joana Margarida Nunes da Costa (University of Coimbra)

Florian Naef (University of Geneva)

Yu Qiao (Shaanxi Normal University)

Hajime Sato (Nagoya University)

[[Martin Schlichenmaier]] (University of Luxembourg)

Yunhe Sheng (Jilin University)

Song Gao (Peking University)

[[Mathieu Stienon]] (Penn State University)

Bingjun Su (Jilin University)

Shanzhong Sun (Capital Normal University)

Xiang Tang (Washington University in St. Louis)

[[Fridrich Valach]] (University of Geneva)

Luca Vitagliano  (Universita degli Studi di Salerno & Istituto Nazionale di Fisica Nucleare)

Aissa Wade (Penn State University)

Lifang Wang (Henan University)

Hong Wang (Nankai University)

Qi Wang (Jilin University)

Ke Wu (Capital Normal University)

Zhiwei Wu (Ningbo University)

Yirui Xiong (Sichuan University)

[[Ping Xu]] (Penn State University)

Xiaomeng Xu (University of Geneva)

Song Yang (Sichuan University)

Xiangdong Yang (Sichuan University)

Yijun Yao (Fudan University)

Yanbin Yin (Henan University)

Tao Zhang (Henan Normal University)

[[Weiping Zhang]] (Nankai University)

Suo Zhao (Sichuan University)

Jieheng Zeng (Sichuan University)

Deshou Zhong (China Youth University of Political Studies)

[[Chenchang Zhu]]  (Gottingen University)

Nguyen Tien Zung (Institut de Mathematiques de Toulouse)

\*-to be confirmed

***

## Trivial Information##

[ Trivial Information](http://www.nim.nankai.edu.cn/nim_e/travel.htm)

***

###Schedule ###

|        |            |        |
|-----|----|-----|
| Time | Speaker|Title|
|  Mon, Jan 4  | | |
|08:45-09:25|Yvette Kosmann-Schwarzbach|On double Poisson brackets|
|09:35-10:15|Xiang Tang|Integration of exact Courant algebroids and the Liu-Weinstein-Xu 2-groupoids|
|10:45-11:25|Aissa Wade|On Cosymplectic-Njenhuis Groupoids|
|11:35-12:15|Martin Schlichenmaier|Some naturally defined star products for Kähler manifolds|
|15:00-15:40|Si Li|BV quantization and algebraic index|
|15:50-16:30|Kentaro Mikami|Cohomology groups of homogeneous Poisson structures|
|17:00-17:40|Hong Wang|Poisson Reductions of Controlled Hamiltonian System with Symmetry|
|  Tue, Jan 5  | | |
|08:45-09:25|Samuel Evens|Degenerations of cohomology rings|
|09:35-10:15|Huijun Fan|Analytic Torsion of singularities|
|10:45-11:25|Mathieu Stienon|Infinite jets of exponential maps|
|11:35-12:15|Camille Laurent-Gengoux|Singular foliations and Lie infinity structures|
|15:00-15:40|Anton Alekseev|Moduli of flat connection and multiplicity spaces|
|15:50-16:30|Nguyen Tien Zung|Action-angle variables on contact manifolds|
|17:00-17:40|Chenchang Zhu|String principal bundles and Courant algebroids|
|  Wen, Jan 6  | | |
|08:45-09:25|Kirill Mackenzie|The outlines of double Lie theory|
|09:35-10:15|Jianghua Lu|On a Poisson structure on products of flag varieties|
|10:45-11:25|Zhiwei Wu|New Hierarchies of Derivative nonlinear Schr\"{o}dinger Type Equations|
|  Thu, Jan 7  | | |
|08:45-09:25|Xiaomeng Xu|Quantum groups via Stokes phenomenon|
|09:35-10:15|Francesco Bonechi|Quantization of Poisson manifolds from multiplicative integrable models|
|15:50-16:30|Hajime Sato|Samuelson condition for bi-Lagrangian structure|
|10:45-11:25|Hui Li|The fundamental group of $G$-manifolds|
|15:00-15:40|Ping Xu|Symplectic realizations of holomorphic Poisson manifolds|
|15:50-16:30|Zhuo Chen|Holomorphic Poisson Structures and its Cohomology on Nilmanifolds|
|  Fri, Jan 8  | | |
|08:45-09:25|Joana Nunes da Costa|Some geometric structures on pre-Courant algebroids|
|09:35-10:15|Luca Vitagliano|Dirac-Jacobi Bundles|
|10:45-11:25|Shanzhong Sun|Double Poisson algebra structures on Fukaya categories|
|11:35-12:15|Janusz Grabowski|Weighted groupoids and algebroids with applications|


Poisson holography

#Contents#
* table of contents
{:toc}

## Idea

A version of the [[holographic principle]] relating particle-mechanics controlled by [[Poisson manifold]]-[[phase spaces]] to [[2d Chern-Simons theory]] in the form of the [[Poisson sigma-model]] 2d [[field theory]].

See the discussion and references here:

* [holographic principle -- Poisson holography](holographic+principle#PoissonHolography)

  * _[off-shell Poisson bracket -- boundary field theory interpretation](off-shell+Poisson+bracket#BoundaryFieldTheoryInterpretation)_

  * _[motivic quantization -- The Poisson manifold at the boundary of 2d Poisson-Chern-Simons theory](motivic%20quantization#PoissonManifoldAtTheBoundaryOf2dChernSimonsTheory)_

## Further literature

* [[Joost Nuiten]], Sec. 5.2.2 in: *[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]*  (August 2013)

which is one of a couple of running examples in

* *[[schreiber:Quantization via Linear homotopy types]]*

Alternative discussion via [[factorization algebras]]:

* [[Eugene Rabinovich]], *A Classical Bulk-Boundary Correspondence* [[arXiv:2202.12332](https://arxiv.org/abs/2202.12332)]


Poisson Lie 2-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A $(X, \omega)$ a [[n-plectic manifold|2-plectic manifold]], its _Poisson Lie 2-algebra_ is the higher analog of the [[Poisson bracket]] [[Lie algebra]] of a [[symplectic manifold]].

For the moment see [here](n-plectic+geometry#PoissonLInfinityAlgebras) for more.

## References

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))
 {#Rogers}


Poisson Lie 2-group

#Contents#
* table of contents
{:toc}

## Idea

The notion of [[Poisson Lie group]] in [[Lie 2-group]] theory.

## References

* [[Zhuo Chen]], [[Mathieu Stiénon]], [[Ping Xu]], _Poisson 2-groups_ ([arXiv](https://arxiv.org/abs/1202.0079))

[[!redirects Poisson Lie 2-groups]]

[[!redirects Poisson 2-group]]
[[!redirects Poisson 2-groups]]


Poisson Lie algebroid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

A Poisson Lie algebroid on a [[manifold]] $X$ is a [[Lie algebroid]]
on $X$ naturally defined from and defining the structure of a
[[Poisson manifold]] on $X$.

This is the degree-1 example of a tower of related concepts, described
at [[n-symplectic manifold]].

## Definition ##

Let $\pi \in \Gamma(\Wedge^2 TX)$ be a [[Poisson manifold]] structure, incarnated as a [[Poisson tensor]].

### As vector-bundle with anchor

In terms of the vector-bundle-with anchor definition of [[Lie
algebroid]] the **Poisson Lie algebroid** $\mathfrak{P}(X,\pi)$ corresponding to $\pi$ is the [[cotangent bundle]]

$$
  \array{
      T^* X &&\stackrel{\pi(-)}{\to}&& T X
      \\
      & \searrow && \swarrow
      \\
      && X
  }
$$

equipped with the anchor map that sends a [[differential 1-form]] $\alpha$ to the [[vector]] obtained by contraction with the [[Poisson bivector]] $\pi \colon \alpha \mapsto \pi(\alpha,-)$.

The [[Lie bracket]] $[-,-] : \Gamma(T^* X) \wedge \Gamma(T^* X) \to \Gamma(T^* X)$ is given by

$$
 [\alpha,\beta]
   \coloneqq
  \mathcal{L}_{\pi(\alpha)} \beta
  - \mathcal{L}_{\pi(\beta)} \alpha
  - d_{dR}(\pi(\alpha,\beta))\,,
$$

where $\mathcal{L}$ denotes the [[Lie derivative]] and $d_{dR}$ the [[de Rham differential]]. This is the unique Lie algebroid bracket on $T^* X \stackrel{\pi}{\to} T X$ which is given on exact differential 1-forms by

$$
  [d_{dR} f, d_{dR} g] = d_{dR} \{f,g\}
$$

for all $f,g \in C^\infty(X)$. On a [[coordinate patch]] this reduces to

$$
  [d x^i , d x^j] = d_{dR} \pi^{i j}
$$

for $\{x^i\}$ the coordinate functios and $\{\pi^{i j}\}$ the components of the [[Poisson tensor]] in these coordinates.

### Chevalley-Eilenberg algebra

We describe the [[Chevalley-Eilenberg algebra]] of the Poisson Lie algebra given by $\pi$, which defines it dually.

Notice that $\pi$ is an element of degree 2 in the [[exterior algebra]]
 $\wedge^\bullet \Gamma(T X)$ of [[multivector field]]s on $X$. The Lie
 bracket on [[tangent bundle|tangent vector]]s in $\Gamma(T X)$
 extends to a bracket $[-,-]_{Sch}$ on multivector field, the
 **[[Schouten bracket]]**. The defining property of the Poisson structure
 $\pi$ is that

$$
  [\pi,\pi]_{Sch} = 0
  \,.
$$

This makes

$$
  d_{CE(\mathfrak{P}(X,\pi))} := [\pi, -] : CE(\mathfrak{P}(X,\pi)) \to CE(\mathfrak{P}(X,\pi)))
$$

into a differential of degree +1 on multivector fields, that squares to 0. We write $CE(\mathfrak{P}(X,\pi))$ for the exterior algebra equipped with this
differential.

More explicitly, let $\{x^i\} : X \to \mathbb{R}^{dim X}$ be a coordinate patch. Then the differential of $CE(\mathfrak{P}(X,\pi))$ is given by

$$
  d_{\mathfrak{P}(X,\pi)} : x^i \mapsto  2 \pi^{i j} \partial_j
$$

$$
  d_{\mathfrak{P}(X,\pi)} : \partial_i \mapsto ...
  \,.
$$

## Properties

### Cohomology and Chern-Simons elements {#Cohomology}

We discuss aspects of the [[∞-Lie algebra cohomology|∞-Lie algebroid cohomology]] of Poisson Lie algebroids $\mathfrak{P}(X,\pi)$. This is equivalently called _[[Poisson cohomology]]_ (see there for details).

We shall be lazy (and follow tradition) and write the following formulas in a local coordinate patch $\{x^i\}$ for $X$.

Then the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{P}(X,\pi))$ is generated from the $x^i$ and the $\partial_i$, and the [[Weil algebra]] $W(\mathfrak{P}(X,\pi))$ is generated from $x^i$, $\partial_i$ and their shifted partners, which we shall write $\mathbf{d} x^i$ and $\mathbf{d}\partial_i$. The differential on the Weil algebra we may then write

$$
  d_{W(\mathfrak{P}(X,\pi))} = [\pi,-]_{Sch} + \mathbf{d}
  \,.
$$

Notice that $\pi \in CE(\mathfrak{P}(X,\pi))$ is a [[∞-Lie algebra cohomology|Lie algebroid cocycle]], since

$$
  d_{CE(\mathfrak{P}(X,\pi))} \pi = [\pi,\pi]_{Sch} = 0
  \,.
$$

+-- {: .num_prop}
###### Proposition

The [[invariant polynomial]] in transgression with $\pi$ is

$$
  \omega =  (\mathbf{d}\partial_i) \wedge (\mathbf{d}x^i)
   \in
  W(\mathfrak{P}(X,\pi))
  \,.
$$

=--

+-- {: .proof}
###### Proof

One checks that the following is a **[[Chern-Simons element]]** (see there for more) exhibiting the transgression

$$
  cs_\pi = \pi^{i j} \partial_i  \wedge \partial_j
  + \partial_i \wedge \mathbf{d}x^i
  \;\;\;
  \in W(\mathfrak{P}(X,\pi))
$$

in that $d_{W(\mathfrak{P}(X,\pi))} cs_\pi = \omega$, and the restriction of $cs_\pi$ to $CE(\mathfrak{P}(X,\pi))$ is evidently the Poisson tensor $\pi$.

For the record (and for the signs) here is the explicit computation

$$
  \begin{aligned}
    d_{W(\mathfrak{P}(X,\pi))} (\pi^{i j} \partial_i \wedge \partial_j
   + \partial_i \wedge \mathbf{d} x^i)
    = &
   \mathbf{d}x^k (\partial_k \pi^{i j}) \partial_i \wedge \partial_j
   \\
   & + 2 \pi^{i j} (\mathbf{d}\partial_i) \wedge \partial_j
   \\
   & - (\partial_i \pi^{j k}) \partial_j \wedge \partial_k \wedge
       \mathbf{d}x^i
   \\
   & +
   (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i)
   \\
   & + (-)(-) 2\pi^{i j} \partial_i \wedge \mathbf{d}\partial_j
   \\
   = &
   (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i)
  \end{aligned}
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

The invariant polynomial $\omega$ makes $\mathfrak{P}(X,\pi)$ a [[schreiber:symplectic ∞-Lie algebroid]].

=--

+-- {: .num_remark}
###### Remark

The [[schreiber:infinity-Chern-Simons theory]] [[action functional]] induced from the above Chern-Simons element is that of the [[Poisson sigma-model]]:

it sends [[∞-Lie algebroid valued forms]]

$$
  \Omega^\bullet(\Sigma) \leftarrow W(\mathfrak{P}(X,\pi))
  (X,\eta)
$$

on a 2-dimensional manifold $\Sigma$ with values in a Poisson Lie algebroid on $X$ to the integral of the [[Chern-Simons form|Chern-Simons 2-form]]

$$
  \Omega^\bullet(\Sigma) \leftarrow
  W(\mathfrak{P}(X,\pi))
  \stackrel{(\omega, cs_\omega)}{\leftarrow}
  W(b^2 \mathbb{R})
  :
  CS_\omega(X,\eta)
$$

which, by the above, is in components

$$
  CS_\omega(X,\eta) = \eta_i \wedge d_{dR} X^i + \pi^{i j} \eta_i \wedge \eta_j
  \,.
$$

=--

### Lagrangian submanifolds and coisotropic submanifolds

The [[Lagrangian dg-submanifolds]] (see there for more) of a Poisson Lie algebroid correspond to the [[coisotropic submanifolds]] of the corresponding [[Poisson manifold]].

## Related concepts ##

* Under [[Lie integration]] a Poisson Lie algebroid is supposed to yield a [[symplectic groupoid]].

* There is a formulation of [[Legendre transformation]] in terms of Lie algebroid.

* [[symplectic Lie n-algebroid]]

  * [[symplectic manifold]]

  * **Poisson Lie algebroid**

  * [[Courant algebroid]]

* [[Hopf algebroid]] (appears as a deformation quantization of a Poisson-Lie algebroid)

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## Examples

* For the simple but important special case of [[Lie-Poisson structure]] see there at _[Poisson-Lie algebroid cohomology](Lie-Poisson+structure#PoissonLieAlgebroidCohomology)_.

## References

One of the earliest reference seems to be

* [[Ted Courant]], _Tangent Lie algebroid_ ([pdf](http://www.iop.org/EJ/article/0305-4470/27/13/026/ja941326.pdf))

A review is for instance in

* [[Dmitry Roytenberg]], appendix A _Courant algebroids, derived brackets and even symplectic supermanifolds_ ([arXiv:math/9910078](http://arxiv.org/abs/math/9910078))

The [[H-cohomology]] of the graded symplectic form of a Poisson Lie algebroid, regarded a a [[symplectic Lie n-algebroid]], is considered in

* {#Severa05} [[Pavol ?evera]], p. 1 of _On the origin of the BV operator on odd symplectic supermanifolds_, Lett Math Phys (2006) 78: 55. ([arXiv:0506331](https://arxiv.org/abs/math/0506331))

[[!redirects Poisson-Lie algebroid]]

[[!redirects Poisson Lie algebroids]]


Poisson Lie group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **Poisson Lie group** (often written Poisson-Lie group) or Lie Poisson group $G$ is a [[ Lie group]] and a [[Poisson manifolds]], the two structures being compatible such that the group product is a smooth map of Poisson manifolds where $G\times G$ has the product Poisson structure.

**Warning**: the inverse map is NOT a Poisson map unless G has the trivial Poisson structure, in fact it is an anti-Poisson map.

[[deformation quantization|Deformation quantizations]] of Poisson Lie groups are [[Hopf algebras]]. The usual [[quantum groups]] have smaller number of quantum subgroups (i.e. Hopf quotient algebras) than the corresponding Lie group has, namely only only those whose classical limits are not only Lie subgroups but Poisson Lie subgroups.

## Properties

### Relation to quantum groups

One can regard Poisson groups as the [[classical limit]] of [[quantum groups]]: a theorem by Drinfeld established a [[bijection]] between connected, simply connected Poisson Lie groups and [[Lie bialgebras]].

## Examples

### Additive Poisson Lie groups
 {#AdditivePoissonLieGroups}

A [[Lie-Poisson structure]] is an additive Poisson Lie group (e.g. [Kosmann-Schwarzbach 04, p. 46](#KosmannSchwarzbach04)).

If $H$ is any (finite dimensional) Lie group then the dual $(T_e H)^*$ of its [[tangent Lie algebra]] has a canonical bracket introduced by Kirillov which makes it into a Poisson Lie group. To this aim one identifies $(T_e H)^*$ with its own tangent space $T_u (T_e H)^*$ and interprets the differential $df$ of a function $f:(T_e H)^*\to \mathbb{R}$ as a function $(T_e H)^*\to ((T_e H)^*)^*\cong T_e H$ where the finite dimensionality is used. Then Kirillov defines

$$
\lbrace f_1, f_2\rbrace (u) = \langle [(df_1)_u, (df_2)_u, u\rangle
$$

Given two Lie groups $H,K$, the Lie algebra homomorphisms $T_e H \to T_e K$ are in 1-1 correspondence with the Poisson Lie maps $(T_e K)^* \to (T_e H)^*$.

## Related concepts

* [[dressing action]]
* [[Poisson Lie 2-group]]

* _Unrelated_ is the concept of the Lie group that [[Lie integration|integrates]] the [[Lie algebra]] inside a [[Poisson algebra]]. This is instead called a _[[quantomorphism group]]_.

## References

* V. Chari, A. Pressley, _A guide to quantum groups_, Camb. Univ. Press 1994
* [[S. Majid]], _Foundations of quantum group theory_, Cambridge University Press 1995, 2000.
* Gloria Marí Beffa, _A transverse structure for the Lie-Poisson bracket on the dual of the Virasoro algebra_,  Pacific J. Math. __163__ (1994),  no. 1, 43--72, [euclid](http://projecteuclid.org/getRecord?id=euclid.pjm/1102622628)
* M. A. Semenov-Tian-Shansky, _Группы Пуассона–Ли. Квантовый принцип двойственности и скрученный квантовый дубль_, Teoret. Mat. Fiz., 1992, 93:2,  302–329 (in Russian) [pdf](http://83.149.209.141/php/getFT.phtml?jrnid=tmf&paperid=1530&volume=93&year=1992&issue=2&fpage=302&what=fullt&option_lang=eng); in English: _Poisson–Lie groups. The quantum duality principle and the twisted quantum double_, Theoret. and Math. Physics 1992, 93:2, 1292–1307 [doi](http://dx.doi.org/10.1007/BF01083527); _Poisson groups and dressing transformations_, Zap. Nauchn. Sem. LOMI, 150 (1986),  119–142 [mathnet.ru](http://mi.mathnet.ru/znsl4975); _Poisson Lie groups, quantum duality principle, and the quantum double_, in: Math. aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992),  219--248, Contemp. Math. __175__, AMS 1994.
* M. A. Semenov-Tian-Shansky, _Classical $r$-matrices, Lax equations, Poisson Lie groups and dressing transformations_, in:  Field theory, quantum gravity and strings, II (Meudon/Paris, 1985/1986), 174--214, Lec. Notes in Phys. __280__, Springer 1987, [MR89g:58098](http://www.ams.org/mathscinet-getitem?mr=89g:58098)
* A. P. Fordy, A. G. Reyman, M. A. Semenov-Tian-Shansky, _Classical $r$-matrices and compatible Poisson brackets for coupled KdV systems_,  Lett. Math. Phys. __17__ (1989), no. 1, 25--29, [doi](http://dx.doi.org/10.1007/BF00420010), [MR90e:58061](http://www.ams.org/mathscinet-getitem?mr=90e:58061)
* A. Yu. Alekseev, A. Z. Malkin, _Symplectic structures associated to Lie-Poisson groups_,  Comm. Math. Phys. __162__  (1994),  no. 1, 147--173.
* T. Tao's blog: [The Euler-Arnold equation](http://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation)
* [[Peter Olver]], _Applications of Lie groups to differential equations_, Springer
* A. Cannas da Silva, [[Alan Weinstein]], _Geometric models for noncommutative algebras_, Berkeley Math. Lec. Notes Series, AMS 1999, [pdf](http://math.berkeley.edu/%7Ealanw/Models.pdf)

* [[Yvette Kosmann-Schwarzbach]], _Groupes de Lie-Poisson quasitriangulaires_, in:  Géométrie symplectique et mécanique (La Grande Motte, 1988),  161--177, Springer LNM __1416__, 1990.

* {#KosmannSchwarzbach04} [[Yvette Kosmann-Schwarzbach]], _Lie bialgebras, Poisson Lie groups and dressing transformations_, in _Integrability of Nonlinear Systems_, Second edition, Lecture Notes in Physics 638, Springer-Verlag, 2004, pp. 107-173. ([pdf](http://www.math.polytechnique.fr/cmat/kosmann/lnp2.pdf))

* A. G. Reyman, _Poisson structures related to quantum groups_, in: Quantum groups and their applications in physics (Varenna, 1994), 407--443,
Proc. Internat. School Phys. Enrico Fermi, 127, IOS, Amsterdam, 1996, [MR97j:58052](http://www.ams.org/mathscinet-getitem?mr=97j:58052)
* Nicola Ciccoli, _Quantization of co-isotropic subgroups_, Lett. Math. Phys. __42__:2 (1997) 123--138, [doi](http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1023/A:1007352218739), [MR98k:58252](http://www.ams.org/mathscinet-getitem?mr=98k:58252)
* Renaud Brahami, _Cluster X-varieties for dual Poisson-Lie groups I, II_, [arxiv/1005.5289](http://arxiv.org/abs/1005.5289), [arxiv/1006.4583](http://arxiv.org/abs/1006.4583)

* László Fehér, [[Ctirad Klimčík]], _Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction_,  Lett. Math. Phys. __87__ (2009), no. 1-2, 125--138, [doi](http://dx.doi.org/10.1007), [MR2010c:53122](http://www.ams.org/mathscinet-getitem?mr=2010c:53122))

* V. Fock, A. B. Goncharov, _Cluster X-varieties, amalgamation and Poisson-Lie groups_, [math.RT/0508408](https://arxiv.org/abs/math/0508408).

* Ivan Calvo, Fernando Falceto, David Garcia-Alvarez, _Topological [[Poisson sigma model]]s on Poisson-Lie groups_, JHEP 0310 (2003) 033, [arXiv](https://arxiv.org/abs/hep-th/0307178).

* Philip Boalch, _[[Stokes phenomenon|Stokes matrices]] and Poisson Lie groups_, Invent. math. 146, 479-506 (2001), [math.DG/0011062](https://arxiv.org/abs/math/0011062).

* [[Ctirad Klimčík]], [[Pavol Ševera]], _[[T-duality]] and the moment map_, IHES/P/96/70, [hep-th/9610198](http://arxiv.org/abs/hep-th/9610198); _Poisson-Lie T-duality: open strings and D-branes_, CERN-TH/95-339. Phys.Lett. B376 (1996) 82-89, [hep-th/9512124](http://arxiv.org/abs/hep-th/9512124)

* [[Anton Alekseev]], [[Ctirad Klimčík]], [[Arkady Tseytlin]], _Quantum Poisson-Lie T-duality and WZNW model_, Nucl. Phys. B458:430-444 (1996) [hep-th/9509123](http://arxiv.org/abs/hep-th/9509.3123)
* David Li-Bland, [[Pavol Ševera]], _On deformation quantization of Poisson-Lie groups and moduli spaces of flat connections_, [arXiv/1307.2047](http://arxiv.org/abs/1307.2047)

[[!redirects Poisson-Lie group]]
[[!redirects Lie Poisson group]]
[[!redirects Lie-Poisson group]]

[[!redirects Poisson Lie groups]]
[[!redirects Poisson-Lie groups]]
[[!redirects Lie Poisson groups]]
[[!redirects Lie-Poisson groups]]

[[!redirects Poisson group]]
[[!redirects Poisson groups]]


Poisson manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Poisson manifolds are a mathematical setup for [[classical mechanics]] with finitely many degrees of freedom.

## Definition

A __[[Poisson algebra]]__ is a commutative unital [[associative algebra]] $A$, in this case over the field of [[real number|real]] or [[complex numbers]], equipped with a [[Lie bracket]] $\{,\} \colon A\otimes A\to A$ such that, for any $f\in A$, $\{ f,-\} \colon A\to A$ is a [[derivation]] of $A$ as an associative algebra.

A __Poisson manifold__ is a real [[smooth manifold]] $M$ equipped with a __Poisson structure__. A Poisson structure is a [[Lie bracket]] $\{,\} \colon C^\infty(M)\times C^\infty(M)\to C^\infty(M)$ on the vector space of [[smooth functions]] on $M$ which together with the pointwise multiplication of functions makes it a Poisson algebra. As derivations of $C^\infty(M)$ [[derivations of smooth functions are vector fields|correspond]] to smooth [[tangent vector fields]], for each $f\in C^\infty(M)$ there is a vector $X_f$ given by $X_f(g)=\{f,g\}$ and called the __[[Hamiltonian vector field]]__ corresponding to the function $f$, which is viewed as a classical hamiltonian function.

Alternatively a Poisson structure on a manifold is given by a choice of smooth antisymmetric [[bivector]] called a __Poisson bivector__ $P\in\Lambda^2 T M$; then $\{f,g\}:=\langle d f\otimes d g, P\rangle$.

This induces and is equivalently encoded by the structure of a [[Poisson Lie algebroid]].

A [[morphism]] $h \colon M\to N$ of Poisson manifolds is a [[smooth function]] between the [[underlying]] [[smooth manifolds]] which preserves the Poisson brackets, in that for all $f, g \in C^\infty(N)$, $\{f\circ h, g\circ h\}_M = \{f,g\}_N$.

## Examples

### Various

+-- {: .num_example}
###### Example

Every manifold admits the _trivial Poisson structure_ for which the [[Poisson bracket]] simply vanishes on all elements.

=--

+-- {: .num_example}
###### Example

Every [[symplectic manifold]] carries a natural Poisson structure. Given a smooth function $f: X \to \mathbb{R}$, the symplectic form $\omega$ induces [[Hamiltonian vector field]] $H_f: X \to T X$. We can then define the Poisson bracket as
$$
  \{f, g\} = H_f g = \omega(H_f, H_g).
$$
See [below](PresymplecticManifolds) for more. However, such Poisson manifolds are very special. It is a basic theorem that Poisson structures on a manifold are equivalent to the smooth [[foliations]] of the underlying manifold such that each leaf is a symplectic manifold.

=--

+-- {: .num_example}
###### Example

The dual to a finite-dimensional [[Lie algebra]] has a natural structure of a Poisson manifold, the _[[Lie-Poisson structure]]_. Its leaves are called [[coadjoint orbits]].

=--

+-- {: .num_example}
###### Example

Given a [[symplectic manifold]] $(X,\omega)$ and given a [[Hamiltonian]] [[function]] $H \colon X \longrightarrow \mathbb{R}$, there is a Poisson bracket on the functions on the [[smooth space|smooth]] [[path space]] $[I,X]$ (the "space of histories" or "space of [[trajectories]]"), for $I = [0,1]$ the closed [[interval]], which is such that its [[symplectic leaves]] are each a copy of $X$, but regarded as the space of initial conditions for evolution with respect to $H$ with a [[source]] term added. For more on this see at _[[off-shell Poisson bracket]]_.

=--

+-- {: .num_example}
###### Example

Every [[local action functional]] which admits a [[Green's function]] for its [[equations of motion]] defines the [[Peierls bracket]] on [[covariant phase space]] (where in fact it is symplectic) and also "[[off-shell Poisson bracket|off-shell]]" on all of configuration space, where it is a genuine Poisson bracket, the canonocal Poisson bracket of the corresponding [[prequantum field theory]].

=--

### Pre-symplectic manifolds and infinitesimal quantomorphisms
 {#PresymplecticManifolds}

We discuss the traditional definition of the [[Poisson bracket]] of a ([[presymplectic manifold|pre-]])[[symplectic manifold]] $(X,\omega)$, and then show how it may equivalently be understood as the algebra of infinitesimal symmetries of any of the [[prequantizations]] of $(X,\omega)$. For more on this see at _[[Poisson bracket Lie n-algebra]]_ and at _[[geometry of physics -- prequantum geometry]]_.

+-- {: .num_defn #PresymplecticManifold}
###### Definition

Let $X$ be a [[smooth manifold]]. A closed [[differential 2-form]] $\omega \in \Omega_{cl}^2(X)$ is a _[[symplectic form]]_ if it is non-degenerate in that the [[kernel]] of the operation of contracting with [[vector fields]]

$$
  \iota_{(-)}\omega \colon Vect(X) \longrightarrow \Omega^1(X)
$$

is trivial: $ker(\iota_{(-)}\omega) = 0$.

If $\omega$ is just closed with possibly non-trivial kernel, we call it a [[presymplectic form]]. (We do not require here the dimension of the kernel restricted to each tangent space to be constant.)

=--

+-- {: .num_defn #HamiltonianVectorField}
###### Definition

Given a [[presymplectic manifold]] $(X, \omega)$, then a _[[Hamiltonian]]_ for a [[vector field]] $v \in Vect(X)$ is a [[smooth function]] $H \in C^\infty(X)$ such that

$$
  \iota_{v} \omega + d H = 0
  \,.
$$

If $v \in Vect(X)$ is such that there _exists_ at least one Hamiltonian for it then it is called a _[[Hamiltonian vector field]]_. Write

$$
  HamVect(X,\omega) \hookrightarrow Vect(X)
$$

for the $\mathbb{R}$-[[linear subspace]] of Hamiltonian vector fields among all [[vector fields]]

=--

+-- {: .num_remark}
###### Remark

When $\omega$ is symplectic then, evidently, there is a unique Hamiltonian vector field, def. \ref{HamiltonianVectorField}, associated with every Hamiltonian, i.e. every smooth function is then the Hamiltonian of precisely one Hamiltonian vector field (but two different Hamiltonians may still have the same Hamiltonian vector field uniquely associated with them). As far as [[prequantum geometry]] is concerned, this is all that the non-degeneracy condition that makes a closed 2-form be symplectic is for. But we will see that the definitions of [[Poisson brackets]] and of [[quantomorphism groups]] directly generalize also to the presymplectic situation, simply by considering not just Hamiltonian functions but pairs of a Hamiltonian vector field and a compatible Hamiltonian.

=--

+-- {: .num_defn #PoissonBracket}
###### Definition

Let $(X,\omega)$ be a [[presymplectic manifold]]. Write

$$
  Ham(X,\omega) \hookrightarrow HamVect(X,\omega) \oplus C^\infty(X)
$$

for the [[linear subspace]] of the [[direct sum]] of [[Hamiltonian vector fields]], def. \ref{HamiltonianVectorField}, and [[smooth functions]] on those pairs $(v,H)$ for which $H$ is a [[Hamiltonian]] for $v$

$$
  Ham(X,\omega)
  \coloneqq
  \left\{
    (v,H) | \iota_v \omega + d H  = 0
  \right\}
  \,.
$$

Define a [[bilinear map]]

$$
  [-,-] \;\colon\;
  Ham(X,\omega) \otimes Ham(X,\omega)
  \longrightarrow
  Ham(X,\omega)
$$

by

$$
 [(v_1,H_1), (v_2,H_2)] \coloneqq ([v_1,v_2], \iota_{v_2}\iota_{v_1} \omega)
 \,,
$$

called the _[[Poisson bracket]]_, where $[v_1,v_2]$ is the standard Lie bracket on [[vector fields]]. Write

$$
  \mathfrak{Pois}(X,\omega)
  \coloneqq
  (Ham(X,\omega),[-,-])
$$

for the resulting [[Lie algebra]]. In the case that $\omega$ is symplectic, then $Ham(X,\omega) \simeq C^\infty(X)$ and hence in this case

$$
  \mathfrak{Pois}(X,\omega)
  \simeq
  (C^\infty(X),[-,-])
  \,.
$$

=--

+-- {: .num_example #HeisenbergAlgebraOfSymplecticVectorSpace}
###### Example

Let $X = \mathbb{R}^{2n}$ and let $\omega = \sum_{i = 1}^n d p_i \wedge d q^i$ for $\{q^i\}_{i = 1}^n$ the [[canonical coordinates]] on one copy of $\mathbb{R}^n$ and $\{p_i\}_{i = 1}^n$ that on the other ("[[canonical momenta]]"). Hence let $(X,\omega)$ be a [[symplectic vector space]] of dimension $2n$, regarded as a [[symplectic manifold]].

Then $Vect(X)$ is [[linear span|spanned]] over $C^\infty(X)$ by the canonical bases vector fields $\{\partial_{q^i}, \partial_{p^i}\}$. These basis vector fields are manifestly [[Hamiltonian vector fields]] via

$$
  \iota_{\partial_{q^i}} \omega = - d p_i
$$

$$
  \iota_{\partial_{p_i}} \omega = + d q^i
  \,.
$$

Moreover, since $X$ is [[connected topological space|connected]], these Hamiltonians are unique up to a choice of constant function. Write $\mathbf{i} \in C^\infty(X)$ for the unit constant function, then the nontrivial Poisson brackets between the basis vector fields are

$$
  [q^i, p_j]
  \coloneqq
  [(-\partial_{p_i}, q^i), (\partial_{q^j}, p_j)]
  =
  -
  \delta_j^i
  (0, \mathbf{i})
  =
  -
  \delta_j^i
  \mathbf{i}
  \,.
$$

This is called the [[Heisenberg algebra]].

More generally, the [[Hamiltonian vector fields]] corresponding to [[quadratic Hamiltonians]], i.e. degree-2 [[polynomials]] in the $\{q^i\}$ and $\{p_i\}$, generate the [[affine symplectic group]] of $(X,\omega)$. The freedom to add constant terms to Hamiltonians gives the [[extended affine symplectic group]].

=--

Example \ref{HeisenbergAlgebraOfSymplecticVectorSpace} serves to motivate a more conceptual origin of the definition of the Poisson bracket in def. \ref{PoissonBracket}.

+-- {: .num_example }
###### Example

Write

$$
  \theta \coloneqq \sum_{i = 1}^n p_i d q_i \in \Omega^1(\mathbb{R}^{2n})
$$

for the canonical choice of [[differential 1-form]] satisfying

$$
  d \theta = \omega
  \,.
$$

If we regard $\mathbb{R}^{2n} \simeq T^\ast \mathbb{R}^n$ as the [[cotangent bundle]] of the [[Cartesian space]] $\mathbb{R}^n$, then this is what is known as the [[Liouville-Poincaré 1-form]].

Since $\mathbb{R}^{2n}$ is [[contractible topological space|contractible]] as a [[topological space]], every [[circle bundle]] over it is necessarily trivial, and hence any choice of 1-form such as $\theta$ may canonically be thought of as being a [[connection on a bundle|connection]] on the trivial $U(1)$-[[principal bundle]]. As such this $\theta$ is a [[prequantization]] of $(\mathbb{R}^{2n}, \sum_{i=1}^n d p_i \wedge d q^i)$.

Being thus a [[circle bundle with connection]], $\theta$ has more symmetry than its [[curvature]] $\omega$ has: for $\alpha \in C^\infty(\mathbb{R}^{2n}, U(1))$ any smooth function, then

$$
  \theta \mapsto \theta + d\alpha
$$

is the [[gauge transformation]] of $\theta$, leading to a different but equivalent [[prequantization]] of $\omega$.

Hence while a [[vector field]] $v$ is said to preserve $\omega$ (is a [[symplectic vector field]]) if the [[Lie derivative]] of $\omega$ along $v$ vanishes

$$
  \mathcal{L}_v \omega = 0
$$

in the presence of a choice for $\theta$ the right condition to ask for is that there is $\alpha$ such that

$$
  \mathcal{L}_v \theta = d \alpha
  \,.
$$

=--

For more on this see also at _[[prequantized Lagrangian correspondence]]_.

Notice then the following basic but important fact.

+-- {: .num_prop}
###### Proposition

For $(X,\omega)$ a [[presymplectic manifold]] and $\theta \in \Omega^1(X)$ a 1-form such that $d \theta = \omega$ then for $(v,\alpha) \in Vect(X)\oplus C^\infty(X)$ the condition

$$
  \mathcal{L}_v \theta = d \alpha
$$

is equivalent to the condition that makes $H \coloneqq \iota_v \theta - \alpha $ a [[Hamiltonian]] for $v$ according to def. \ref{HamiltonianVectorField}:

$$
  \iota_v \omega + d (\iota_v \theta - \alpha  ) = 0
  \,.
$$

Moreover, the [[Poisson bracket]], def. \ref{PoissonBracket}, between two such Hamiltonian pairs $(v_i, \alpha_i -\iota_v \theta)$ is equivalently given by the skew-symmetric [[Lie derivative]] of the corresponding vector fields on the $\alpha_i$:

\[
  \label{EquationForLieHomomorphism}
  \iota_{[v_1,v_2]} \theta
  -
  \iota_{v_2}\iota_{v_1}\omega
  =
  \mathcal{L}_{v_1} \alpha_2 - \mathcal{L}_{v_2} \alpha_1
\]

=--

+-- {: .proof}
###### Proof

Using [[Cartan's magic formula]] and by the prequantization condition $d \theta = \omega$ the we have

$$
  \begin{aligned}
    \mathcal{L}_v \theta &= \iota_v d\theta + d \iota_v \theta
    \\
    & =  \iota_v\omega + d \iota_v \theta
  \end{aligned}
  \,.
$$

This gives the first statement. For the second we first use the formula for the [[de Rham differential]] and then again the definition of the $\alpha_i$

$$
  \begin{aligned}
    \iota_{v_2}\iota_{v_1} \omega
    & =
    \iota_{v_2}\iota_{v_1} d\theta
    \\
    & =
    \iota_{v_1} d \iota_{v_2} \theta
    -
    \iota_{v_2} d \iota_{v_1} \theta
    -
    \iota_{[v_1,v_2]} \theta
    \\
     & =
    \iota_{v_1} d \alpha_2 - \iota_{v_1} \iota_{v_2}\omega
    - \iota_{v_2} d \alpha_1 + \iota_{v_2} \iota_{v_1}\omega
    -
    \iota_{[v_1,v_2]} \theta
    \\
     & =
    2 \iota_{v_2} \iota_{v_1}\omega
    + \mathcal{L}_{v_1} \alpha_2
     -\mathcal{L}_{v_2} \alpha_1
    -
    \iota_{[v_1,v_2]} \theta
 \end{aligned}
  \,.
$$

=--

+-- {: .num_cor #EquivalenceBetweenPoissonBracketAndInfinQuantomorphism}
###### Corollary

For $(X,\omega)$ a [[presymplectic manifold]] with $\theta \in \Omega^1(X)$ such that $d \theta = \omega$, consider the [[Lie algebra]]

$$
  \mathfrak{QuantMorph}(X,\theta)
  =
  \left\{
    (v,\alpha) | \mathcal{L}_v \theta = d \alpha
  \right\}
  \subset
  Vect(X) \oplus C^\infty(X)
$$

with Lie bracket

$$
  [(v_1,\alpha_1), (v_2,\alpha_2)] = ([v_1,v_2], \mathcal{L}_{v_1}\alpha_2 - \mathcal{L}_{v_2}\alpha_1)
  \,.
$$

Then by (eq:EquationForLieHomomorphism) the linear map

$$
  (v,H) \mapsto (v, \iota_v \theta - H)
$$

is an [[isomorphism]] of [[Lie algebras]]

$$
  \mathfrak{Pois}(X,\omega)
    \stackrel{\simeq}{\longrightarrow}
  \mathfrak{QuantMorph}(X,\theta)
$$

from the [[Poisson bracket Lie algebra]], def. \ref{PoissonBracket}.

=--

This shows that for exact pre-symplectic forms the Poisson bracket Lie algebra is secretly the Lie algebra of infinitesimal symmetries of any of its [[prequantizations]]. In fact this holds true also when the pre-symplectic form is not exact:

+-- {: .num_defn #CechDelignePrequantizationAnditsInfinitesimalAutomorpisms}
###### Definition

For $(X,\omega)$ a [[presymplectic manifold]], a [[Cech cohomology|Cech]]-[[Deligne cohomology|Deligne]] [[cocycle]] $(X,\{U_i\},\{g_{i j}, \theta_i\})$ for a  _[[prequantization]]_ of $(X,\omega)$ is

1. an [[open cover]] $\{U_i \to X\}_i$;

1. 1-forms $\{\theta_i \in \Omega^1(U_i)\}$;

1. smooth function $\{g_{i j} \in C^\infty(U_{i j}, U(1))\}$

such that

1. $d \theta_i = \omega|_{U_i}$ on all $U_i$;

1. $\theta_j = \theta_i + d log g_{ij}$ on all $U_{i j}$;

1. $g_{i j} g_{j k} = g_{i k}$ on all $U_{i j k}$.

The _quantomorphism Lie algebra_ of this is

$$
  \mathfrak{QuantMorph}(X,\{U_i\},\{g_{i j}, \theta_i\})
  =
  \left\{
    (v, \{\alpha_i\})
    |
    \mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i
    \,,
    \mathcal{L}_v \theta_i = d \alpha_i
  \right\}
  \subset
  Vect(X) \oplus \left(\underset{i}{\bigoplus} C^\infty(U_i)\right)
$$

with bracket

$$
  [(v_1, \{(\alpha_1)_i\}), (v_2, \{(\alpha_2)_i\})]
  \coloneqq
  ([v_1,v_2], \{\mathcal{L}_{v_1}(\alpha_2)_i - \mathcal{L}_{v_2} (\alpha_1)_i\})
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

For $(X,\omega)$ a [[presymplectic manifold]] and
$(X,\{U_i\},\{g_{i j}, \theta_i\})$ a [[prequantization]],
def. \ref{CechDelignePrequantizationAnditsInfinitesimalAutomorpisms}, the linear map

$$
  (v,H) \mapsto (v,  \{\iota_v \theta_i - H|_{U_i}\})
$$

constitutes an [[isomorphism]] of [[Lie algebras]]

$$
  \mathfrak{Pois}(X,\omega)
  \stackrel{\simeq}{\longrightarrow}
  \mathfrak{QuantMorph}(X,\{U_i\},\{g_{i j}, \theta_i\})
  \,.
$$

=--

+-- {: .proof}
###### Proof

The condition $\mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i$
on the infinitesimal quantomorphisms, togther with the Cech-Deligne cocycle condition $d log g_{i j} = \theta_j - \theta_i$ says that on $U_{i j}$

$$
  \iota_v \theta_j - \alpha_j = \iota_v \theta_i - \alpha_i
$$

and hence that there is a globally defined function $H \in C^\infty(X)$ such that $\iota_v \theta_i - \alpha_i = H|_{U_i}$. This shows that the map is an isomrophism of vector spaces.

Now over each $U_i$ the the situation for the brackets is just that of corollary
\ref{EquivalenceBetweenPoissonBracketAndInfinQuantomorphism} implied by (eq:EquationForLieHomomorphism), hence the morphism is a Lie homomorphism.

=--

## Properties

### Deformation quantization

[[Kontsevich formality]] implies that every Poisson manifold has a family of [[deformation quantizations]], parameterized by the [[Grothendieck-Teichmüller group]].

## Related concepts

* [[Poisson tensor]]

* [[Poisson geometry]]

* [[Poisson cohomology]]

* [[Poisson reduction]]

* [[isotropic submanifold]], [[coisotropic submanifold]]

* [[symplectic leaf]]

* [[symplectic realization]]

* [[symplectic dual pair]]

* [[Poisson Lie algebroid]]

* [[deformation quantization]]

  * [[Moyal quantization]]

* [[higher Poisson structure]]

  * [[Nambu bracket]]

  * [[Poisson n-algebra]]

  * [[Poisson bracket Lie n-algebra]]

\linebreak

[[!include Isbell duality - table]]

\linebreak

[[!include infinitesimal and local - table]]

\linebreak

## References

The notion is due to:

* [[André Lichnerowicz]], *Les variétés de Poisson et leurs algèbres de Lie associées*, Journal of Differential Geometry **12** 2 (1977) 253–300 [[doi:10.4310/jdg/1214433987](http://dx.doi.org/10.4310/jdg/1214433987)]

Further early discussion:

* [[Alan Weinstein]],  *The local structure of Poisson manifolds*,  J. Differential Geom. **18** 3 (1983) 523-557 [[doi:10.4310/jdg/1214437787](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-18/issue-3/The-local-structure-of-Poisson-manifolds/10.4310/jdg/1214437787.full)]

Monograph on [[Poisson geometry]]:

* [[Marius Crainic]], [[Rui Loja Fernandes]], *Lectures on Poisson Geometry*, Graduate Studies in Mathematics **217**, Amer. Math. Soc. (2021) [[ISBN:978-1-4704-6430-1](https://bookstore.ams.org/view?ProductCode=GSM/217)]

See also:

* Wikipedia, *[Poisson manifold](https://en.wikipedia.org/wiki/Poisson_manifold)*

Discussion in [[Lagrangian field theory]] includes:

* {#BeniniSchenkel16} [[Marco Benini]], [[Alexander Schenkel]], _Poisson algebras for non-linear field theories in the Cahiers topos_, Annales Henri Poincar\'e, 18(4), 1435-1464 (2017) ([arXiv:1602.00708](https://arxiv.org/abs/1602.00708))

[[!redirects Poisson Lie algebra]]

[[!redirects Poisson manifolds]]

[[!redirects Poisson bracket]]
[[!redirects Poisson brackets]]

[[!redirects Poisson structure]]
[[!redirects Poisson structures]]

[[!redirects Poisson Lie algebra]]
[[!redirects Poisson Lie algebras]]

[[!redirects Poisson bracket Lie algebra]]
[[!redirects Poisson bracket Lie algebras]]


Poisson n-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $n \in \mathbb{N}$,  a **Poisson $n$-algebra** $A$ is a [[Poisson algebra]] $A$ in a [[category of chain complexes]] with Poisson bracket of degree $(1-n)$ (which is a bracket of degree 0 on $\mathbf{B}^{n-1} A$).

## Properties

### Relation to $E_n$-algebras
 {#RelationToEnAlgebras}

The [[homology]] of an [[E-n algebra]] for $n \geq 2$ is a Poisson $n$-algebra.

Moreover, in [[chain complexes]] over a [[field]] of [[characteristic zero]]  the [[E-n operad]] is [[formal dg-algebra|formal]] (see _[[the little n-disk operad is formal]]_), hence equivalent to its homology, and so in this context $E_n$-algebras are equivalent to Poisson $n$-algebras.

This fact is a higher analog of [[Kontsevich formality]]. It means that every higher dimensional [[prequantum field theory]] given by a $P_n$ algebra does have a [[deformation quantization]] (as [[factorization algebras]]) and that the space of choice of these a torsor over the [[automorphism infinity-group]] of $E_n$, a higher analog of the [[Grothendieck-Teichmüller group]].

See also tho _[MO discussion](#MODiscussion) linked to below_.

### Relation to $L_\infty$-algebras
 {#RelationToLInfinityAlgebras}

There is a [[forgetful functor]] from Poisson $n$-algebras to [[dg-Lie algebras]] given by forgetting the [[associative algebra]] structure and by [[suspension of a chain complex|shifting]] the underlying chain complex by $(n-1)$.

Conversely, this functor has a [[derived functor|derived]] [[left adjoint]] which sends a [[dg-Lie algebra]] $(\mathfrak{g},d)$ to its _[[universal enveloping E-n algebra|universal enveloping Poisson n-algebra]]_ $(Sym(\mathfrak{g}[n-1], d))$. (See also [Gwilliam, section 4.5](#Gwilliam)).

## Examples

* A Poisson 1-algebra is a [[Poisson algebra]].

* A Poisson 2-algebra is a [[Gerstenhaber algebra]].

* The [[Chevalley-Eilenberg algebra]] of a [[symplectic Lie n-algebroid]] (see there for details) is naturally a Poisson $(1+n)$-algebra.

* A [[classical BV complex]] is naturally (if obtained as a [[derived critical locus]], or else by definition) a Poisson 0-algebra.

## Related concepts

* [[BV-algebra]]

* [[higher Poisson structure]]

  * [[Nambu bracket]]

  * [[Poisson bracket Lie n-algebra]]

[[!include Isbell duality - table]]

[[!include deformation quantization - table]]

## References

* [[Alberto Cattaneo]], [[Domenico Fiorenza]], R. Longoni, _Graded Poisson Algebras_, Encyclopedia of Mathematical Physics, eds. J.-P. Françoise, G.L. Naber and Tsou S.T. , vol. 2, p. 560-567 (Oxford: Elsevier, 2006). ([pdf](http://www.math.uzh.ch/fileadmin/math/preprints/15-05.pdf))

An introduction to Poisson $n$-algebras in [[dg-geometry]]/[[symplectic Lie n-algebroids]] is in section 4.2 of

* [[Alberto Cattaneo]], [[Florian Schätz]], _Introduction to supergeometry_, Lecture notes of a course held at the school Poisson 2010 at IMPA, July 2010; 21  ([arXiv:1011.3401](http://arxiv.org/abs/1011.3401))

For discussion in the context of [[perturbation theory|perturbative]] [[quantum field theory]]/[[factorization algebras]]/[[BV-quantization]] see

* [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in perturbative quantum field theory : $P_0$-operad_ ([wiki](http://math.northwestern.edu/~costello/factorization_public.html#[[P_0%20operad]]), [pdf](http://math.northwestern.edu/~gwilliam/factorization.pdf))

* {#Gwilliam} [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([pdf](http://math.berkeley.edu/~gwilliam/thesis.pdf))

and for further references along these lines see at _[[factorization algebra]]_.

For general discusison of the relation to [[E-n algebras]] see

* MO discussion, _[Poisson algebras as deformations vs. Poisson algebras in algebraic topology](http://mathoverflow.net/questions/74405/poisson-algebras-as-deformations-vs-poisson-algebras-in-algebraic-topology/75194#75194)_
 {#MODiscussion}

[[!redirects Poisson n-algebras]]

[[!redirects Poisson 0-algebra]]
[[!redirects Poisson 1-algebra]]
[[!redirects Poisson 2-algebra]]
[[!redirects Poisson 3-algebra]]

[[!redirects P-n algebra]]
[[!redirects P-0 algebra]]
[[!redirects P-1 algebra]]
[[!redirects P-2 algebra]]

[[!redirects P-n algebras]]
[[!redirects P-0 algebras]]
[[!redirects P-1 algebras]]
[[!redirects P-2 algebras]]

[[!redirects Poisson 0-algebras]]
[[!redirects Poisson 1-algebras]]
[[!redirects Poisson 2-algebras]]
[[!redirects Poisson 3-algebras]]

[[!redirects Pn-algebra]]
[[!redirects Pn-algebras]]


Poisson pencil

If $\{,\}$ and $\{,\}'$ are two Poisson brackets on the same commutative algebra, or on the same manifold, then we say that they are **compatible** if any linear combination of these brackets $a\{,\}+b\{,\}'$ is again a Poisson bracket on the same commutative algebra or manifold. If so, all linear combination make a 2-parametric family of Poisson brackets. Such families are called Poisson pencils.

[[!redirects Poisson pencil]]


Poisson reduction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The generalization of [[symplectic reduction]] from [[symplectic manifolds]] to [[Poisson manifolds]].

## Related concepts

* [[gauge reduction]], [[symplectic reduction]], [[presymplectic reduction]], [[BV-BRST formalism]]

[[!include symplectic reduction - table]]

* [[quantization commutes with reduction]]

## References

* Juan-Pablo Ortega, Tudor S. Ratiu, _Poisson reduction_ ([arXiv:math/0508635](http://arxiv.org/abs/math/0508635))

[[!redirects Poisson reduction]]


Poisson sigma-model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents
* automatic table of contents goes here
{:toc}

## Idea

The _Poisson $\sigma$-model_ is a 2-dimensional [[sigma-model]] [[quantum field theory]] whose target space is a [[Poisson Lie algebroid]]. It is a [[2-dimensional Chern-Simons theory]]. This may be thought of as encoding the [[quantum mechanics]] of a [[string theory|string]] propagating on the [[phase space]] of a system in [[classical mechanics]].

In his solution of the problem of [[deformation quantization]] [[Maxim Kontsevich]] showed that correlators for the 2-string interaction (the correlator on the worldsheet that is a disk with three marked points on its boundary) describe a product operation which is a deformation of the [[Poisson bracket]] on the target space. This solves the formal [[deformation quantization]] problem of the phase space in [[quantum mechanics]] by identifying the quantum algebra with the _open string algebra_ of a [[string theory]] on that target.

The principal variant of the nonlinear Poisson sigma model is sometimes called Cattaneo-Felder model who have shown the graphical expansion used in Kontsevich's approach to the deformation quantization is explained via a Feynman diagram expansion in this model.

If one considers [[brane]]s in the target space of the Poisson sigma-model, then then algebra of open strings that used to be just the deformation of the Poisson algebra becomes an [[A-infinity algebra|A-infinity algebroid]]. (See the references below).

> Probably something close to a [[Calabi-Yau category]], hence identifying the Poisson sigma-model as a [[TCFT]]. Does anyone know more in this direction?

## Definition

The [[target space]] of a Poisson $\sigma$-model is any [[Poisson manifold]] $(X, \{\})$, or rather the [[Poisson Lie algebroid]] $\mathfrak{P}$ corresponding to that.

A field configuration on a 2-dimensional $\Sigma$ is a [[connection on an infinity-bundle|connection]]

$$
  (\phi,\eta) : \mathfrak{T}^*\Sigma \to \mathfrak{P}
 \,.
$$

In components this is

1. a [[smooth function]] $\phi : \Sigma \to X$;

1. a 1-form $\eta \in \Omega^1(\Sigma, \phi^* T^* X)$
   with values in the pullback of the [[cotangent bundle]] of $X$ along $\phi$.

The [[action functional]] on the [[configuration space]] of all such connections for [[compact space|compact]] $\Sigma$ is defined to be

$$
  S : (\phi,\eta) \mapsto
  \int_\Sigma
  \left(
     \langle \eta \wedge d_{dR}\phi\rangle
     +
     \frac{1}{2}
     \phi^*\pi(\eta)
  \right)
  \,,
$$

where $\pi \in \wedge^2_{C^\infty(C)}\Gamma(T X)$ is the Poisson tensor of $(X, \{-,-\})$ and where  $\langle -,-\rangle$ is the canonical [[invariant polynomial]] on the [[Poisson Lie algebroid]].

## Properties

### Relation to deformation quantization of Poisson manifolds
 {#RelationToQuantizationOfPoissonManifolds}

In ([Cattaneo-Felder](#CattaneoFelder)) it was shown that the [[n-point function|3-point function]] in the [[path integral quantization]] of the Poisson $\sigma$-model of a [[Poisson Lie algebroid]] associated with a [[Poisson manifold]]  computes the [[star product]] in the [[deformation quantization]] of the Poisson manifold as given by ([Kontsevich](#Kontsevich)).

A [[higher geometric quantization]] that also yields the [[strict deformation quantization]] is discussed at _[[extended geometric quantization of 2d Chern-Simons theory]]_.

One may think of this relation between the 2d Poisson sigma-model and [[quantum mechanics]] = 1d [[quantum field theory]] as an example of the Chern-Simons type [[holographic principle]]. For more along these lines see below at _[holographic dual](#HolographicDual)_.

### Branes
 {#Branes}

The [[branes]] of the Poisson sigma model are related to
[[coisotropic submanifolds]] of the underlying [[Poisson manifold]]. Notice that these are the [[Lagrangian dg-submanifolds]] of the [[Poisson Lie algebroid]]. ([Cattaneo-Felder 03](#CattaneoFelder03)).

### Holographic dual
 {#HolographicDual}

By the Chern-Simons form of the [[holographic principle]] one expects the Poisson sigma-model to be related to a 1-dimensional [[quantum field theory]]. This is [[quantum mechanics]]. The [above](#RelationToQuantizationOfPoissonManifolds) relation to the deformation quantization of Poisson manifolds goes in this direction. More explicit realizations have been attempted, for instance ([Vassilevich](#Vassilevich)).

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[sigma-model]]

  * [[AKSZ sigma-model]]

    * **Poisson sigma-model**

      * [[A-model]], [[B-model]]

    * [[Courant sigma-model]]

      * [[Chern-Simons theory]]

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * [[4d Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

  * [[infinite-dimensional Chern-Simons theory]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

### General

The Poisson sigma model was first considered in

* [[Noriaki Ikeda]], _Two-dimensional gravity and nonlinear gauge theory_ , Ann.Phys.235(1994) 435- 464, [hep-th/9312059].

and later independently by P. Schaller, T. Strobl, motivated from an attempt to unify several two-dimensional models of [[gravity]] and to cast them into a common form with [[Yang-Mills theory|Yang-Mills theories]].

* P. Schaller, T. Strobl, _Poisson structure induced (topological) field theories_, Modern Phys. Lett. A 9 (1994), no. 33, 3129--3136, [doi](http://dx.doi.org/10.1142/S0217732394002951); _Introduction to Poisson $\sigma$-models_,  Low-dimensional models in statistical physics and quantum field theory (Schladming, 1995),  321--333, Lecture Notes in Phys. __469__, Springer 1996.

* Thomas Strobl, _Gravity from Lie algebroid morphisms_, Comm. Math. Phys.  __246__  (2004),  no. 3, 475--502, _Algebroid Yang-Mills theories_,   Phys. Rev. Lett.  __93__  (2004),  no. 21, 211601, 4 pp. [doi](http://dx.doi.org/10.1103/PhysRevLett.93.211601)

* [[Martin Bojowald]], [[Alexei Kotov]], Thomas Strobl, _Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries_,  J. Geom. Phys.  __54__:4  (2005) 400--426, [doi](https://doi.org/10.1016/j.geomphys.2004.11.002)

* Ctirad Klimčík, T. Strobl, _WZW-Poisson manifolds_, J. Geom. Phys. __43__ (2002), no. 4, 341--344, doi

The detailed argument by Cattaneo and Felder on how [[Maxim Kontsevich]]'s formula for the [[deformation quantization]] star product is the 3-point function of the Poisson sigma-model is in

* {#CattaneoFelder} [[Alberto Cattaneo]], [[Giovanni Felder]], _A path integral approach to the Kontsevich quantization formula_, Commun. Math. Phys. 212, 591--611 (2000) [doi](http://dx.doi.org/10.1007/s002200000229), [math.QA/9902090](http://arxiv.org/abs/math/9902090).

* [[Alberto Cattaneo]], [[Giovanni Felder]], _Poisson sigma models and deformation quantization_, Mod. Phys. Lett. A 16, 179--190 (2001) [hep-th/0102208](http://arxiv.org/abs/hep-th/0102208).

See also

* [[Alberto Cattaneo]], [[Giovanni Felder]], _Poisson sigma models and symplectic groupoids_ , (ed. [[Klaas Landsman]], M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198, 61--93 (Birkhäuser, 2001)
[math.SG/0003023](http://arxiv.org/abs/math/0003023).

* [[Alberto Cattaneo]], [[Giovanni Felder]], _On the AKSZ formulation of the Poisson sigma model_, Lett. Math. Phys. 56, 163--179 (2001) [math.QA/0102108](http://arxiv.org/abs/math/0102108).

The interpretation in terms of [[schreiber:infinity-Chern-Simons theory]] is discussed in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]_ (2011)

Discussion in terms of [[holography]] is in

* D. V. Vassilevich, _Holographic duals to Poisson sigma models_ ([arXiv:1301.7029](http://arxiv.org/abs/1301.7029))
 {#Vassilevich}

### With branes
 {#LitWithBranes}

The study of [[branes]] in the Poisson sigma-model has been started in

* Damien Calaque, [[Giovanni Felder]], Andrea Ferrario, Carlo A. Rossi, _Bimodules and branes in deformation quantization_ ([arXiv:0908.2299](http://arxiv.org/abs/0908.2299))

* Damien Calaque, [[Giovanni Felder]], Carlo A. Rossi, _Deformation quantization with generators and relations_ ([arXiv:0911.4377](http://arxiv.org/abs/0911.4377))

* [[Alberto Cattaneo]], [[Giovanni Felder]], _Coisotropic submanifolds in Poisson geometry and branes in the Poisson $\sigma$-model_, Lett.Math.Phys. 69 (2004) 157-175 ([arXiv:0309180](http://arxiv.org/abs/math/0309180))
 {#CattaneoFelder03}

* Andrea Ferrario, _Poisson Sigma Model with branes and hyperelliptic Riemann surfaces_ ([arXiv:0709.0635](http://arxiv.org/abs/0709.0635))

A review is in

* F. Falceto, _Branes in Poisson sigma models_ (2009) ([pdf](http://benasque.org/2009gph/talks_contr/094Falceto.pdf))

### Recent developments

* Francesco Bonechi, [[Alberto Cattaneo]], [[Pavel Mnev]], _The Poisson sigma model on closed surfaces_ ([arXiv:1110.4850](http://arxiv.org/abs/1110.4850))

[[!redirects Poisson sigma model]]


Poisson summation formula

## Idea

A certain identity for certain [[Fourier transforms]], equating a sum or integral of a function over a domain (e.g., a lattice) with a corresponding sum or integral of its Fourier dual over a dual domain (e.g., the dual lattice).

## Statement

The Poisson summation formula is basic to harmonic analysis over general locally compact Hausdorff abelian groups.

Consider an exact sequence in $TopAb$ of [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological abelian group|abelian groups]]

$$0 \to A \to B \to C \to 0.$$

where $A, B, C$ are equipped with Haar measures $d\mu_A, d\mu_B, d\mu_C$ that make the following equation true:

$$\int_B f(b)\; d\mu_B(b) = \int_C \int_A f(a + c)\; d\mu_A(a) d\mu_C(c)$$

for all [[continuous functions]] $f: B \to \mathbb{C}$ with [[compact support]].
(The inner integral on the right is a shorthand for $\int_A f(a + b)\; d\mu_A(a)$ for any $b \in B$ that maps to $c \in C$; this is well-defined since the integral is invariant under changes $b \mapsto b + a'$ within the same coset $c$.) We remark that given Haar measures $d\mu_A, d\mu_B$, there exists a Haar measure $d\mu_C$ making this Fubini-type equation true. Then, since Haar measures form a torsor over the group of positive reals with multiplication, it follows that any two of $d\mu_A, d\mu_B, d\mu_C$ determine the third.

In this notation, the "Poisson summation formula" is the equation asserted by the following result.

+-- {: .num_theorem}
###### Theorem
Let $\widehat{C}$ denote the [[Pontryagin duality|Pontryagin dual]] of $C$, and $d\mu_{\widehat{C}}$ the dual Haar measure. For any [[Schwartz-Bruhat function]] $f: B \to \mathbb{C}$, we have

$$\int_A f(a)\; d\mu_A = \int_{\widehat{C}} \widehat{f}(\widehat{c})\; d\mu_{\widehat{C}}$$

where $\widehat{f}$ is the Fourier transform of $f$, as a function on $\widehat{B}$ defined by
$$\widehat{f}(\chi) = \int_B f(b)\overline{\chi(b)}d \mu_B(b).$$
=--

In the special case of a [[lattice (discrete subgroup)|lattice]] $L$ inside $B$, the dual space $L^\perp = \widehat{B/L}$ is a lattice inside $\widehat{B}$, and the integrals are over discrete spaces, i.e. integration is just summation and we have

$$\sum_{x \in L} f(x) = \frac1{\mu(B/L)} \sum_{y \in L^\perp} \widehat{f}(y)$$

where $\mu$ is the Haar measure on $B/L$ (as above). Often the measure on $B$ is chosen so that $\mu(B/L) = 1$.

The classical case is when $B$ is a [[Euclidean space]] $\mathbb{R}^n$. But another case of a lattice inside locally compact abelian groups occurs in the context of [Tate's thesis](#Tate1950), where a [[global field]] is viewed as a lattice inside its [[ring of adeles]].

## Examples

* The Poisson summation formulation implies the [[functional equation]] for the [[Jacobi theta function]], which in turn implies that of the [[Riemann zeta function]], see at _[Riemann zeta function -- functional identity](Riemann+zeta+function#functional_equation)_.

## Related entries

* [[functional equation]]

* A nonabelian version of the Poisson summation formula is the [[Selberg trace formula]].

## References

Reviews include

* theorem 4.1 in _Analytic theory of modular forms_ [pdf](http://www.math.harvard.edu/~jbland/ma259x_notes.pdf)

* E. Kowalski, prop. 2.2.1 in  _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))

An application to [[zeta functions]] via harmonic analysis on [[adele rings]] originates in Tate's thesis:

* {#Tate1950} [[John Tate]], _[[Fourier analysis in number fields, and Hecke's zeta-functions]]_, Princeton, May 1950, thesis; reproduced in _Algebraic Number Theory_ (Proc. Instructional Conf., Brighton, 1965) pp. 305–347, Academic Press 1967
[MR0217026](http://www.ams.org/mathscinet-getitem?mr=0217026)

A textbook account is

* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

and brief review in

* {#Garrett11} [[Paul Garrett]], _Iwasawa-Tate on ζ-functions and L-functions_, 2011 ([pdf](http://www-users.math.umn.edu/~garrett/m/mfms/notes_c/Iwasawa-Tate.pdf)
[[!redirects Poisson formula]]