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.
123
questions
75
votes
10
answers
17k
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 ...
24
votes
3
answers
3k
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 ...
21
votes
1
answer
1k
views
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 ...
17
votes
1
answer
2k
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 ...
16
votes
3
answers
1k
views
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 fields ...
15
votes
1
answer
924
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 ...
14
votes
5
answers
2k
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 ...
14
votes
3
answers
2k
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 ...
12
votes
2
answers
888
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 ...
11
votes
2
answers
535
views
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 ...
10
votes
3
answers
2k
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 ...
10
votes
1
answer
404
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 g}\...
9
votes
3
answers
470
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 ...
8
votes
1
answer
897
views
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:
$$
...
7
votes
2
answers
576
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 $k[h]/...
7
votes
0
answers
137
views
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}}\...
6
votes
1
answer
651
views
Bracket systems (generalization of Poisson brackets)
Related to Why symplectic geometry gives Poisson geometry by coming at it from the other side.
This isn't as fully formalized as it probably should be, but I think enough of the idea is there to ask ...
6
votes
1
answer
360
views
2-shifted Poisson bracket on Lie algebra cohomology
Let $\frak{g}$ be a semisimple Lie algebra, and let $({-},{-})$ be an invariant inner product on $\frak{g}$. The Chevalley–Eilenberg complex $C^*(\frak{g})$ has a natural Poisson bracket of degree $-2$...
6
votes
1
answer
1k
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 $...
6
votes
1
answer
842
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 ...
6
votes
0
answers
199
views
Deformation quantization of infinite dimensional Poisson manifolds
In 1999, Karaali wrote a review of formal deformation quantization for a class she took with Weinstein.
She ends the paper with the following remark:
Another question that remains involves the ...
6
votes
0
answers
203
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 ...
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$ ...
5
votes
2
answers
620
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 ...
5
votes
1
answer
449
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 $\...
5
votes
1
answer
284
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 ...
5
votes
1
answer
352
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 g\vert^{1/...
5
votes
2
answers
567
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
0
answers
201
views
Which symplectic manifolds are coadjoint orbits of finite-dimensional $G$ on $\mathfrak{g}^*$ with the Kostant–Kirillov–Souriau Poisson structure?
Given some finite-dimensional symplectic manifold $(M,\omega)$, can we answer whether or not it arises as the coadjoint orbit of some finite-dimensional $G$ acting on the dual $\mathfrak{g}^*$ to its ...
5
votes
0
answers
110
views
Lie groupoids as symmetries of mechanical systems?
Lie groups are well studied as symmetries of mechanical systems in symplectic/Poisson geometry. For instance, if $G$ acts freely and properly on a mechanical system modeled by a symplectic manifold $(...
5
votes
0
answers
240
views
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 ...
5
votes
0
answers
372
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 ...
4
votes
2
answers
472
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
203
views
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 ...
4
votes
1
answer
544
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 ...
4
votes
1
answer
137
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 ...
4
votes
0
answers
115
views
A Poisson structure induced by double Poisson bracket
$\DeclareMathOperator\Sym{Sym}$Let $k$ be a field of characteristic zero. In Van den Bergh's paper Double Poisson algebras, it is shown that a double Poisson bracket on an unital associative algebra $...
4
votes
0
answers
153
views
Definition of the derivative of a Poisson structure on a manifold given by bivector called a Poisson bivector
What is the derivative of a Poisson structure on a manifold given by a Poisson bivector?
4
votes
0
answers
119
views
Star product on functions of a Poisson-Lie group
Consider a Poisson-Lie group $G$, with whatever additional requirements (quasi-triangular, compact, simply connected).
We can consider $G$ as a Poisson Manifold and apply Kontsevich formality to ...
4
votes
0
answers
216
views
Deformation of the Hochschild-Kostant-Rosenberg isomorphism for universal enveloping algebra
Let $\mathfrak{g}$ be a Lie algebra over a char. $0$ field and let $\iota: U\mathfrak{g}\rightarrow S\mathfrak{g}$ be the Poincaré-Birkhoff-Witt (PBW) isomorphism, inverse to that natural map from the ...
4
votes
0
answers
138
views
"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 ...
4
votes
0
answers
233
views
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 ...
4
votes
0
answers
217
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 $\mu:P\...
3
votes
3
answers
857
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 ...
3
votes
1
answer
649
views
Courant algebroids from Poisson geometry
Could somebody provide some simple examples of Courant algebroids coming from Poisson geometry?
3
votes
1
answer
244
views
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 ...
3
votes
1
answer
137
views
Nonstandard Podles spheres as $U_c(\frak{h})$ invariants
In this paper Podles introduced a $2$-parameter family of $q$-deformed spheres $S_{q,c}$ that are now called the "Podles spheres". The case of $c=0$ is very special and is known as the "...
3
votes
1
answer
265
views
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 ...
3
votes
1
answer
300
views
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 ...
3
votes
3
answers
258
views
Nontrivial Poisson relations for affine Poisson algebras
Let $A$ be a polynomial algebra over a field of characteristic $0$ in the variables $x_1,\dots,x_n$. Consider polynomials $f_1,\dots,f_m\in A$ and let $I$ be the ideal they generate in $A$. Moreover, ...