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Questions tagged [poisson-geometry]

Poisson geometry is the study of varieties endowed with a Poisson structure, which is a certain kind of 2-tensor. This is closely related to symplectic geometry.

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Interpretation of the Schouten bracket as an integrability condition

The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $\Lambda$ the vanishing of the Schouten bracket $[\Lambda,\Lambda]=0$ is referred to as a sort ...
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If $A$ is a (shifted) Poisson algebra, what does $A[\varepsilon]$ represent?

I have a question which is not really precise, unfortunately. Let $A$ be a Poisson $n$-algebra, i.e. a graded commutative algebra with a Lie bracket of degree $n-1$ s.t. the bracket is a biderivation ...
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Is there a classification of polynomial Poisson brackets?

As an example, consider the following Poisson bracket on ${\mathbb R}^n$: $$\{x_i, x_{i+1}\} = x_ix_{i+1}(x_i+x_{i+1}),\\ \{x_i, x_{i+2}\} = x_ix_{i+1}x_{i+2}.$$ The indices are taken modulo $n$, and ...
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intersection of Voronoi cell and Circle

My question is about overlapping a random Voronoi cell and a circle. Suppose there are some Poisson Voronoi cells generated by a homogeneous Poisson Point Process with density λ and Voronoi ...
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“Signature Changing” Generalization of Lie Algebra?

I have in mind a mathematical structure I've never heard of before. Does anyone know what might be? It is a manifold with vector fields whose Lie brackets have structure coefficients that are ...
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Decompose elements in $SL_2$ as a pair of elements in $SL_2^*$.

I have a question about decomposing elements in $SL_2$ as a pair of elements in $SL_2^*$. Here $SL_2^*$ is the dual Poisson Lie group of $SL_2$ which is defined as follows. Let $G$ be a Poisson-Lie ...
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How to show that the structure constant on $\mathcal{G}^*$ is $C_{c}^{ab} = f_{cd}^b r^{ad} + f_{cd}^a r^{db}$?

Let $(\mathcal{G}, \mathcal{G}^*, \delta)$ be a Lie bialgebra. Suppose that the structure constant on $\mathcal{G}^*$ and $\mathcal{G}$ are \begin{align} & [t^a, t^b]_* = C_c^{ab} t_c, \\ & [...
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115 views

Trying to understand dressing actions

I am reading the lecture notes and trying to understand dressing actions. Let $G$ be a Poisson-Lie group and $G^*$ its dual Poisson-Lie group. In the lecture notes above, Proposition 5.22 on page 80,...
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Is the action $T \times G \to G$ Poisson?

Let $G$ be a Poisson-Lie group. Let $M$ be a symplectic manifold. In the paper, the third paragraph of page 1238, it is said that an action $G \times M \to M$ is called Poisson if $G \times M \to M$ ...
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Reference request for poisson group actions which are not hamiltonian

Hamiltonian Lie group actions of Poisson manifolds are well studied and found everywhere in literature. I am wondering if there is any material available on what is known about Poisson actions in ...
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Symplectic submanifolds of cotangent bundles of Lie groups

So, my question specifically pertains to $T^*SO(3)$ but I guess adjusted it could be asked about Lie groups in general. The canonical symplectic form on the cotangent bundle is invariant under the ...
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Poisson cohomology of germfied Poisson structures in dimension two

Let $f(x, y)$ be a smooth function in the real case or a holomorphic function in the complex case. Denote $\pi=f(x, y)\frac{\partial}{\partial x}\wedge \frac{\partial}{\partial y}$ be the ...
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Could we extend the star product on a Poisson manifold from its ring of smooth functions to its de Rham complex?

Let $M$ be a smooth manifold with a Poisson bracket $\{-,-\}$. Kontsevich proved that there exists a deformation quantization of $M$, i.e. let $C^{\infty}(M)[[\hbar]]=C^{\infty}(M)\otimes_{\mathbb{R}}\...
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Is every singular foliation induced by a Lie algebroid?

Let $M$ be a smooth manifold. A smooth distribution $D$ on $M$ is the union of a family $\{D_p \leq T_p M : p\in M\}$ of vector spaces such that there is a family $\mathcal C $ of smooth vector ...
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Lagrangian submanifold of Poisson manifolds

Let $V$ be a finite dimensional vector space. Let $\psi\in \Lambda^2V$ be a (possibly degenerate) $2$-vector. Then $\psi$ defines a map $V^*\rightarrow V$. Let $U\subset V$ denote the image of this ...
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Symplectic reduction and canonical one form

Let $X$ be a smooth manifold. Then $T^*X$ is canonically a symplectic manifold. Let $\alpha$ denote the canonical one form on $T^*X$ (so in local coordinates, $\alpha=\sum p_i dq_i$). Then the ...
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How to compute the deformation quantizations of a polynomial Poisson algebra?

As a specialist told me, given a smooth affine Poisson algebra $S$ over $\mathbb{C}$, up to a choice of certain characteristic class, one can find one of the deformation quantizations of $S$, say $A$, ...
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How to prove a bracket is super anti-commutative?

On page 12 of the paper, there is a formula about super Poisson bracket on a Lie super group $G$: \begin{align} \{\phi, \psi\} = \sum_{\mu, \nu} (-1)^{|\phi||\nu|} r^{\mu \nu} ( R_{\mu} \phi R_{\nu} \...
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Are simple Poisson $A$-modules finitely generated as $A$-modules?

Let $A$ be a Poisson $\mathbb{C}$-algebra: $A$ has a the structure of a complex commutative algebra and at the same time it carries the structure of a Lie algebra, with Lie bracket $\{\cdot, \cdot\}$. ...
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Construct super Poisson brackets on the coordinate rings of Lie super groups

On line 7 of page 61 of the book a guide to quantum groups, a Poisson bracket is defined on $\mathbb{C}[GL_n]$ for every classical $r$-matrix as follows. Let $V$ be a vector space with a basis $v_1, \...
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Super version of Poisson brackets of tensor products

Let $A$ be a Poisson super algebra ($A$ is a super algebra and $A$ satisfies super Jacobi identity, super commutativity, super Leibniz rule). Super version of the product of two tensor products is \...
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Why symplectic geometry gives Poisson geometry

One way I've learned to understand Poisson geometry is to consider it as symplectic geometry with no open conditions - i.e. no condition of nondegeneracy. This idea can be applied to many other ...
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Two definitions of the super Jacobi identity

In this paper, page 149, the super Jacobi identity is given by \begin{align} J(x, y,z) := (-1)^{|x||z|}[[x, y],z] +(-1)^{|z||y|}[[z,x], y]+(-1)^{|y||x|}[[y,z],x] = 0. \end{align} But in this article, ...
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Tensor product of two Poisson modules

Let $H$ be a Poisson algebra. A Poisson $H$-module is a vector space $V$ with two bilinear maps \begin{align} H \otimes V \to V \\ (h,v) \mapsto h.v, \end{align} \begin{align} H \otimes V \to V \\ (h,...
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Poisson cohomology

Let $\Pi$ be a Poisson structure on a manifold $M$. Then we can define a differential $d$ on the complex $\Lambda^{\bullet}M$ $$ C^{\infty}M \to TM \to...\Lambda^kTM \to... $$ in the following way: $$ ...
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156 views

Finite-dimensional Poisson cohomologies

Suppose we have a poisson manifold $M$ whose Poisson cohomology is finite-dimensional in each degree? Does it mean that our manifold is symplectic? Are there other cohomological criteria of ...
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246 views

Poisson structure on the dual Lie algebroid

Let $E \to X$ be a Lie algebroid over the manifold $X$. Let $x_1,...x_n$ be local coordinates on $X$ and $e_1,...e_m$ be the basis of local sections of $E$. In terms of these coordinate functions Lie ...
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Cluster algebra structure compatible with Poisson brackets

Let $X$ be a Poisson variety. There is a concept "cluster algebra structure compatible with Poisson structure" introduced in the paper. Suppose that we construct a maximal independent set of ...
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297 views

Some elementary questions about deformation quantization

I am interested in deformations of affine Poisson algebras, and so this is the setting in which I shall write out the elementary definitions involved. All algebras and vector spaces shall be over $\...
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coisotropic submanifolds on poisson manifolds

Let $(M, \{.,.\})$ be a Poissonmanifold and $B$ the corresponding Poissontensor. Now in this context, a embedded submanifold $C \subset M$ is called coisotropic, if $B^\#(TC^\circ) \subset TC$. For ...
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Symplectic leaves in positive characteristic

I am currently considering a family of filtered algebras over an algebraically closed field of positive characteristic with the property that the associated graded algebra is a finitely generated ...
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Is the antipode anti-bracketed?

In the book Algebras of Functions on Quantum Groups: Part I, Remark 3.1.4, we have the following result. Let $A$ be a Poisson Hopf algebra. That is, $A$ is both a Hopf algebra and a Poisson algebra ...
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Is there a name for a noncommutative generalization of Poisson algebra?

Is there a name for an associative algebra which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c], \hspace{15mm}(*)$$ ...
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Do Kähler realizations of symplectic manifolds exist?

If $(S, \omega)$ is a smooth (not necessarily analytic!) symplectic manifold, does there exist a (almost-)Kähler manifold $K$ and a surjective Poisson submersion $\pi : K \to S$? Think of $\Bbb R ^{...
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Closing the commutative diagram for symplectic realizations

Let $f: (M_1, P_1) \to (M_2, P_2)$ be a Poisson map between Poisson manifolds. Let $\pi_i : (S_i, \omega_i) \to (M_i, P_i), \ i=1,2$ be symplectic realizations. Putting these objects in a rectangular ...
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Recover Poisson bracket on $C[G]$ using the Lie cobracket $\delta: g \to \Lambda^2 g$

By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on ...
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Classifications of Lie bialgebras

What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl_n$ be the general linear Lie algebra. Classify all Lie cobrackets $\...
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1answer
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Glueing together functions defined on the leaves of a foliation

Even though my question can be asked for very general types of foliations, I am interesetd only in its answer for Poisson manifolds, which are what I am currently studying. Consider a Poisson ...
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1answer
101 views

Nash-type theorems for Poisson manifolds

My question comes as a natural follow-up of the previous one which concerned symplectic manifolds: if $(M, P)$ is a Poisson manifold, what embedding theorems are there into some target space (I am ...
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Analytification of Poisson structures on an affine variety

It is well known that one can transfer every affine variety $X$ over $\mathbb{C}$ into an analytic space $X^{an}$ in a natural way. This process is called the analytification. My question is that does ...
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1answer
279 views

Are symplectic realizations of a Poisson manifold unique?

If $(M, P)$ is a (Hausdorff) Poisson manifold, then there exist a surjective Poisson submersion $\pi : (S, \omega) \to (M, P)$ with $(S, \omega)$ a symplectic manifold. I am in a situation where I ...
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1answer
270 views

Questions about Sklyanin bracket

For every classical r-matrix $r$, there is a Poisson bracket called Sklyanin bracket associated to $r$. It is defined in (3.3) of page 5 in (http://arxiv.org/pdf/1101.0015v2.pdf) as follows. \begin{...
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How to solve this system of equations? [closed]

I am trying to find a Poisson bracket on an algebra, and need to find a solution to a system of equations. The system of equations is very complicated, with more than 10000 equations and 60 variables. ...
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119 views

Natural Poisson brackets on $S(V^*)$

Let $V$ be a finite dimensional vector space and $S(V)$ the corresponding symmetric algebra. Suppose that we have a Poisson bracket $\lambda = \{,\}: S(V) \otimes S(V) \to S(V)$. Let $V^*$ be the dual ...
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Is the preimage of coisotropic submanifold coisotropic?

Let $M$ and $N$ be smooth manifolds with Poisson-structures $\{ \cdot , \cdot\}|_M$ and $\{\cdot , \cdot \}|_N$ We call $\phi: M \to N$. a Poisson-map, if the pullback of $\phi$ is compatible with the ...
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Poisson Manifold Structures on Even Dimensional Spheres

The $2n$-sphere, for $n=1,2,3$, possess a (non-trivial) Poisson manifold structure. Is this still true for $n > 3$? Describing the spheres as homogeneous spaces $SO(n)/S(n-1)$, are there Poisson ...
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Classification of finite dimensional Lie subalgebras of $\mathbb R[q^1,\dots,q^n,p_1,\dots,p_n]$

Do there exist results towards answering the following question? Consider the Poisson algebra of regular functions $A=\mathbb R[V]$ on the symplectic vector space $V:=T^* \mathbb R^n$. Using ...
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Kontsevich's flow on the space of Poisson structures

In §5.3 of Kontsevich's Formality Conjecture he writes: This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is ...
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When does a symmetric Poisson manifold decompose into homogeneous pieces?

When people study representations, they pay a lot of attention to the irreducible ones. This is justified by the fact that, roughly speaking, every unitary representation decomposes into a direct ...
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Associativity of Kontsevich's star product up to second order

In Deformation quantization of Poisson manifolds, Kontsevich gives the quantization formula $$f \star g = \sum_{n=0}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma,\alpha}(f,g).$$ He gives ...