# 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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### algebraic momentum map

Let $T$ be a linear algebraic torus over $\mathbb C$ and $X$ be a smooth quasi-projective symplectic $T$-variety. Also, assume that the action of $T $ is free and $X/T$ exists as a smooth variety. Is ...

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### Quantum orbit method at roots of unity

Chari and Pressley’s A guide to quantum groups, or the original work by Vaksman and Soibelman in 1989, explains that, similarly to the orbit method which relates quantised coadjoint orbits to unitary ...

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### 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 $...

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### An analogue of the Poisson bracket in contact geometry?

I was looking at this old question and thought it might get more attention at this site. In summary, the OP asks the following question:
McDuff and Salamon define an analogue of the Poisson bracket ...

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### Are the odd dimensional spheres Poisson homogeneous spaces?

Are the odd dimensional spheres $S^{2n+1}$, for $n \in \mathbb{N}_{\geq 1}$, Poisson homogeneous spaces in the sense of Drinfeld?

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### Does the notion of a Poisson monad exist?

Starting with a monoidal category with duals $C$, one may consider the category $End(C)$ of endofunctors of $C$. A Hopf monad on $C$ is a bimonad on $C$ with (a generalised notion of the) antipode. ...

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### Degeneration of spectral sequence computing Hochschild cohomology of enveloping algebra of Lie algebroid

Let $L$ be a Lie algebroid on a smooth affine $k$-scheme $X=spec(R)$. Recall that by definition $L$ is a locally free sheaf with the structure of a sheaf of $k$ Lie algebras, so that there exists a ...

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### 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 ...

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### Poisson reduction in odd/graded Poisson geometry?

I would like to know whether there is any literature on Poisson reduction of $\mathbb Z$- or $\mathbb Z_2$-graded Poisson algebras.
A $\mathbb Z$-graded Poisson algebra with degree $p\in\mathbb Z$ ...

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### Fixed point scheme definition

I'm sorry if this is a trivial question, but it seems I can't find a clear answer.
I have a finitely generated Poisson algebra $A$, the Poisson scheme $X=Spec(A)$ and an automorphism $g$.
What is ...

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### 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, ...

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### 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?

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### Free almost commutative vertex algebras

Given a commutative $k$-algebra $A$, we can freely generate a commutative vertex algebra by formally adjoining a derivation. We obtain a functor $CAlg_{k} \rightarrow CVAlg_{k} $, which I'll denote $\...

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### 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 ...

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### Lie bracket on the complex valued functions of the space of representations of a Riemann surface

Let $S$ be a closed surface and $G$ be a reductive Lie group. Goldman (Invariant functions on Lie groups and Hamiltonian flows of surface group representations) proved that, for a fairly general class ...

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### Recovering the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids

How can I recover the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids? I am using the formulation of the theorem given in
Rui Loja Fernandes, ...

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### Failure of the Jacobi identity

So I'm facing a problem of physical origin which I'd like to understand on a geometric background.
I have a long, tedious bivector involving functional derivatives. I write what it would be the ...

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### 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 ...

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### 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 ...

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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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### 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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### 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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### 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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### 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 ...