**0**

votes

**0**answers

50 views

### Try to understand the relation between Poisson Lie groups and Lie bialgebras

I am trying to understand the relation between Poisson Lie groups and Lie bialgebras. In the book Lectures on quantum groups, Page 20, Theorem 22 says that given a Poisson Lie group, we can construct ...

**1**

vote

**1**answer

53 views

### 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 ...

**3**

votes

**0**answers

83 views

### 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.,
$$[a, b \circ c]=[a,b]\circ c+b\circ [a,c], \hspace{15mm}(*)$$ ...

**2**

votes

**0**answers

117 views

### 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 ...

**0**

votes

**1**answer

38 views

### 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 ...

**2**

votes

**2**answers

165 views

### 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 ...

**2**

votes

**1**answer

146 views

### 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 ...

**4**

votes

**1**answer

107 views

### 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 ...

**1**

vote

**1**answer

77 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 ...

**0**

votes

**0**answers

79 views

### 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 ...

**1**

vote

**1**answer

103 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 ...

**1**

vote

**1**answer

110 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.
...

**0**

votes

**1**answer

172 views

### 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.
...

**0**

votes

**1**answer

97 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 ...

**1**

vote

**0**answers

42 views

### 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 ...

**2**

votes

**0**answers

125 views

### 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 ...

**1**

vote

**1**answer

78 views

### 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 ...

**15**

votes

**1**answer

855 views

### 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 ...

**2**

votes

**0**answers

31 views

### 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 ...

**11**

votes

**1**answer

638 views

### 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 ...

**1**

vote

**1**answer

224 views

### Integrating Poisson groups

Recall that a symplectic groupoid (http://projecteuclid.org/euclid.bams/1183553676) is a Lie groupoid $\mathcal{G}\rightrightarrows X$ together with a symplectic structure on $\mathcal{G}$ ...

**0**

votes

**0**answers

96 views

### Lie bialgebras cohomology

I am wondering if there exist a universal construction of (co)-chain complex associated to a given algebra to study deformations as Hochschild homology for associative algebras, or Chevalley-Eilenberg ...

**3**

votes

**0**answers

93 views

### Casimirs of Poisson brackets obtained via Poisson reduction

Suppose that a Lie group $G$ acts freely and properly on a symplectic manifold $P$ via symplectomorphisms. Suppose further that we have at our disposal an $\text{Ad}^*$-equivariant momentum map ...

**5**

votes

**2**answers

162 views

### Poisson ideals vs. ideals generated by Poisson central elements

Let $R$ be a Poisson algebra (over $\mathbb C$, say) with Poisson center $Z = \{c \in R : \{c,R\} = 0\}$ and consider two types of ideals $I \leq R$:
$I = \langle (c_i) \rangle$ is generated by ...

**5**

votes

**2**answers

375 views

### Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space?

It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more ...

**2**

votes

**2**answers

269 views

### What is twisted in a twisted Poisson structure?

The usual definition of a twisted Poisson algebra involving a 3-form
does not seem to refer to a Poisson algebra that is then twisted,
i.e. twisting either the commutative product or the Lie bracket.
...

**5**

votes

**1**answer

178 views

### Absent 2nd order terms in deformation quantization of Poisson manifolds

I am reading Kontsevich' famous paper on deformation quantization of Poisson manifolds. In section 1.4.2 on page 4 he gives the general formula for the star product associated to a Poisson structure ...

**3**

votes

**0**answers

183 views

### Hypersurfaces with Gorenstein singular loci

Recall that a hypersurface $D$ in a complex manifold $X$ is called a free divisor if the Lie algebroid $\mathcal{T}_X(-\log D)$ of vector fields tangent to $D$ is locally free. This condition is ...

**4**

votes

**1**answer

365 views

### Poisson structure on the cotangent bundle

Let $X$ be a smooth variety over $\mathbb{C}$ and $\mathscr{A}$ a sheaf of twisted differential operators on $X$. The latter comes equipped with a natural filtration and the associated graded algebra ...

**5**

votes

**1**answer

285 views

### Riemannian and symplectic structures

Let $(\mathcal M,g)$ be a smooth Riemannian manifold and $\Delta$ be the standard (positive) Laplace operator given in coordinates by the usual
$$
\Delta=-\vert g\vert^{-1/2}\partial_j(\vert ...

**9**

votes

**3**answers

376 views

### twisted Poisson structures, degenerate metrics and integrability properties of (2,0)-tensors

Given a regular (constant rank) bi-vector $\Pi \in \Gamma(\bigwedge^2TM)$ on a smooth manifold $M$ the necessary and sufficient condition for the image of $\Pi^\sharp:T^*M\to TM$ to be an integrable ...

**1**

vote

**1**answer

186 views

### Convergence for a family of poisson structures

I would like to get some references. I hope that somebody helps.
Let $(M,\Pi)$ be a smooth Poisson manifold. Let $\delta:\mathcal{V}^{.}(M)\to \mathcal{V}^{.}(M)$ be a differential operator on the ...

**2**

votes

**1**answer

160 views

### Flatness of contravariant connections

In the classical case of covariant connections, the flatness of a connection means that, locally, one has parallel frames around any point. Now, given a flat contravariant connection $\mathcal{D}$ on ...

**5**

votes

**0**answers

300 views

### “Natural” Poisson structure on $(\mathbb{P}^1)^N$

Recently there is some interest in the Poisson geometry of "cluster manifolds", which are varieties associated to cluster algebras. See for example the works of Gekhtman, Shapiro and Vainshtein. In ...

**6**

votes

**0**answers

158 views

### Poisson Ind-Varieties

I am looking for any places in the literature where an author has had occasion to consider Poisson structures on infinite-dimensional algebro-geometric objects, e.g. ind-varieties or proalgebraic ...

**12**

votes

**3**answers

1k views

### Poisson algebras as deformations vs. Poisson algebras in algebraic topology

Commutative Poisson algebras $A$ can be thought of as commutative algebras equipped with a first-order deformation into a noncommutative algebra given by the Poisson bracket. A simple example is the ...

**4**

votes

**0**answers

258 views

### What is the dual bialgebra structure in this special case?

Hi,
I would like to study a special case of Lie bialgebras. Let $(\mathcal{G},<,>)$ a Lie algebra endowed with a scalar product $<,>$ such that
$$\mathcal{G}=S\oplus D(\mathcal{G}),$$
...

**5**

votes

**1**answer

546 views

### How can I see the “missing” Poisson center when the rank of the Poisson structure drops?

Recall that a Poisson algebra is a commutative algebra $A$ along with a bracket $\lbrace,\rbrace: A^{\otimes 2} \to A$ which is a Lie bracket and which is also a derivation in each variable. The ...

**10**

votes

**2**answers

548 views

### What reasonable choices of morphisms are there for the category of Poisson algebras?

The first definition of the category of Poisson algebras that comes to mind is that a morphism between Poisson algebras is an algebra homomorphism that is also a Lie algebra homomorphism with respect ...

**1**

vote

**0**answers

140 views

### What are the Poisson tensors for which hamiltonians are left invariant?

Hi!
Given a Poisson tensor $\pi$ on a Lie group $G$. The hamiltonian $X_f$ associated to a smooth function $f\in C^\infty(G)$ is definied by
$$X_f=-[\pi,f]$$
where $[\,,\,]$ is the Schouten bracket
...

**1**

vote

**1**answer

407 views

### What are dressing transformations, in the context of Poisson-Lie groups?

Hello!
I have some background in Poisson geometry, in particular Poisson-Lie groups and I would like to initiate myself to dressing transformations.
If $(G,\pi)$ is a Poisson-Lie group, then its Lie ...

**2**

votes

**2**answers

257 views

### Hamiltonians of compatible Poisson tensors

Hi!
Given two Poisson tensors $\pi_0$ and $\pi$ (on a Lie group $G$) ; if the two tensors are compatible, i.e.
$$[\pi_0,\pi]=0$$
what are the relations between their hamiltonians ?
If we denote by ...

**6**

votes

**2**answers

454 views

### Which commutative algebras admit a nonzero Poisson bracket?

Let $A$ be a commutative algebra, not necessarily unital, over a field $k$ (of characteristic not equal to $2$, or even equal to $0$, if it helps). A second-order formal deformation of $A$ is a ...

**11**

votes

**5**answers

1k views

### Quantization and noncommutative deformations

Hello,
I would like to introduce myself to the theory of quantization and noncommutative deformations of Riemann Poisson structures. In fact, I am familiar with Riemannian and Poisson geometry, but I ...

**2**

votes

**1**answer

465 views

### Courant algebroids from Poisson geometry

Hello!
Could somebody provide some simple examples of Courant algebroids coming from Poisson geometry?
Thanks!

**7**

votes

**3**answers

925 views

### In the dictionary between Poisson and Quantum, what corresponds to Coisotropic?

I work entirely over a field of characteristic $0$, in case it matters.
Recall that a Poisson algebra is a commutative algebra $A$ with a bracket $\lbrace,\rbrace : A^{\wedge 2} \to A$ which is (1) a ...

**4**

votes

**1**answer

410 views

### Holomorphic Poisson brackets on Fano manifolds

I am looking for the preprint
A. Bondal, Noncommutative
deformations and Poisson brackets on
projective spaces. Preprint MPI/93-67
which I could not find online. Does anyone have an ...

**3**

votes

**3**answers

729 views

### Is there any relation between deformation and extension of Lie algebras?

In a paper of A. Weinstein on the geometry of Poisson manifolds, he relates the formal linearization around a zero, p, of the Poisson bivector to extensions of the Lie algebra induced by the bivector ...

**34**

votes

**8**answers

6k views

### What is a Lagrangian submanifold intuitively?

What are good ways to think about Lagrangian submanifolds?
Why should one care about them?
More generally: same questions about (co)isotropic ones.
Answers from a classical mechanics point of view ...

**5**

votes

**3**answers

2k views

### Differential Hochschild Cohomology, general tools?

Background: in deformation quantization one wants to construct formal associative deformations (the star products $\star$) of the algebra of smooth complex-valued functions on a Poisson manifold $M$ ...