**2**

votes

**0**answers

106 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

67 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

779 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

24 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 ...

**10**

votes

**1**answer

580 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

199 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

78 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

75 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

157 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

318 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

245 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.
...

**4**

votes

**1**answer

165 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

167 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

331 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

273 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

369 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

181 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

140 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 ...

**4**

votes

**0**answers

271 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

147 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

946 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

253 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

508 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

523 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

337 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

256 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

423 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

435 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

866 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

380 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

705 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 ...

**32**

votes

**8**answers

5k 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

1k 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 $M$ ...

**-1**

votes

**1**answer

293 views

### Quantum cohomology of isomorphic Poisson varieties

This question is related with my previous one Quantum cohomology rings as invariants, but now, I want to ask a more concrete thing. If $X$ and $Y$ are Poisson varieties which are isomorphic (as a ...

**10**

votes

**1**answer

308 views

### Is there much theory of superalgebras acting on manifolds by alternating polyvector fields?

Usual story: vector fields on $M$, with their Lie bracket, form a Lie algebra. We can consider "actions" of some other Lie algebra ${\mathfrak g}$ on $M$ by looking at Lie homomorphisms ${\mathfrak ...

**19**

votes

**3**answers

2k views

### Classical mechanics motivation for poisson manifolds?

Suppose I want to understand classical mechanics.
Why should I be interested in arbitrary poisson manifolds and not just in symplectic ones?
What are examples of systems best described by non ...