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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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}}\...
Zhaoting Wei's user avatar
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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 ...
Daniel Teixeira's user avatar
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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 ...
Harold Williams's user avatar
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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 ...
duetosymmetry's user avatar
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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 $(...
user2002's user avatar
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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 ...
Lewis Topley's user avatar
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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 ...
Xin Nie's user avatar
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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 $...
Yining Zhang's user avatar
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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?
Jim Stasheff's user avatar
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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 ...
Rik Voorhaar's user avatar
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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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"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 ...
Lydia Marie Williamson's user avatar
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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 ...
Dr. Evil's user avatar
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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\...
Josh Burby's user avatar
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Particular Lie bialgebra structure

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra, and let $r\in\bigwedge^2\mathfrak{g}$ be a solution of the Yang-Baxter equation. The Yang–Baxter equation states that for all $x\in\...
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Has anyone written down an approach to the Lenard-Magri integrability scheme via algebraic geometry?

I’ve been thinking about the algebro-geometric meaning of the Lenard-Magri scheme of getting an integrable system from a pair of compatible Poisson structures. I think one might be able to prove a ...
rrrrrttttttt's user avatar
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Reference of general version of the PBW theorem and its consequences

Let $A$ be a commutative ring with identity and $L$ be a Lie algebra which is also a free module over $A$. I have seen the following statements: The universal enveloping algebra $U(L)$ is isomorphic (...
Cusp's user avatar
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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 ...
Wodahs's user avatar
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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. ...
Christos's user avatar
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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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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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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, ...
Murilo Luiz's user avatar
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176 views

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 ...
CristinaSardon's user avatar
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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 ^{...
Alex M.'s user avatar
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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 ...
Ago Szekeres's user avatar
3 votes
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331 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 ...
Brent Pym's user avatar
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Nonlinear Poisson brackets associated with nilpotent (matrix) Lie algebras?

With every finite-dimensional Lie algebra $\mathfrak{g}$ one can associate a linear Poisson bracket on $\mathfrak{g}^\ast$. With some more restrictions on $\mathfrak{g}$ and some extra ingredients, ...
Ricardo Buring's user avatar
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114 views

Cohomology theory for Dirac manifolds

I am trying to see if there is any existing cohomology theory for Dirac manifolds. For the case of poisson manifolds, we have the notion of Poisson cohomology. For a manifold $M$, one can consider the ...
Praphulla Koushik's user avatar
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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 ...
Dr. Evil's user avatar
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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 ...
Vectornaut's user avatar
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2 votes
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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 ...
user56980's user avatar
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Special class of bi-hamiltonian systems

A bi-Hamiltonian manifold is a manifold $M$ equipped with two compatible Poisson tensors $\pi_0$ and $\pi$. I am interested in the case of a Lie group $G$ endowed with a multiplicatif Poisson tensor $...
amine's user avatar
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Poisson bracket on $T^*T\mathrm{SU}(1,1)$

Consider the cotangent bundle of the tangent bundle $T^*TG$ of a Lie group $G$. Denote its the Lie algebra by $\mathfrak{g}$. By left translations, we have the trivialization $T^*G \cong G \times \...
Koundinya Vajjha's user avatar
1 vote
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117 views

Integral expression for the Poisson bracket

I already asked this in the physics forum but without much attention, so I thought it might attract more attention here. Is there an integral expression for the Poisson bracket that can be derived ...
Nicolas Medina Sanchez's user avatar
1 vote
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27 views

Connected components of Isotropy types as strata of Poisson leaves

Let $X$ be a smooth affine variety with an algebraic symplectic form $\omega$. Let $G$ be a finite subgroup of the group of symplectomorphisms of $X$. We can say that $X$ is trivially a normal variety ...
Flavius Aetius's user avatar
1 vote
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Differential of tensor product of maps

Let $G$ be a Poisson-Lie group with Poisson bivector field $\pi.$ Let $\pi^{R}$ be the trivialization of $\pi$ with respect to the right translations i.e. $$\pi^{R} (g) = (d_{g} R_{g^{-1}} \otimes d_{...
Anil Bagchi.'s user avatar
1 vote
0 answers
61 views

Is the Lie bracket on $\mathfrak g^{\ast}$ induced from a cocommutator defined on $\mathfrak g\ $?

Let $G$ be a Poisson-Lie group. Let $\mathfrak g = \text {Lie} (G) = T_1 G$ be the corresponding Lie algebra. Then the Poisson structure on $G$ gives rise to a Lie bracket $[\cdot, \cdot]$ on $\...
Anil Bagchi.'s user avatar
1 vote
0 answers
80 views

Integrability of the characteristic distribution of almost Dirac structures

Let $L$ be an almost Dirac structure having an integrable characteristic distribution. What can we say about the involutivity of $L$ under the Courant Bracket? or under which extra conditions can we ...
user2002's user avatar
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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 $p\in\mathbb Z$ ...
AlexArvanitakis's user avatar
1 vote
0 answers
73 views

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 ...
Jianrong Li's user avatar
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1 vote
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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 ...
R Mary's user avatar
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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 ...
motivique's user avatar
1 vote
0 answers
130 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 ...
G.-S. Zhou's user avatar
1 vote
0 answers
87 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 ...
Olorin's user avatar
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0 votes
0 answers
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Concrete examples of quantum duality principle

Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal ...
yohei ohta's user avatar
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0 answers
65 views

Why is $\Pi_r^L$ a non-degenerate Poisson structure on $G\ $?

Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$...
Anil Bagchi.'s user avatar
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0 answers
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Problem in understanding the proof of cocycle condition for cocommutator

Let $G$ be a Poisson–Lie group with Poisson bivector field $\pi$. Let $\pi^{R} \colon G \longrightarrow \bigwedge^2 \mathfrak{g}$ be defined by $$\pi^R (x) = (d_x R_{x^{-1}} \otimes d_x R_{x^{-1}}) \...
Anil Bagchi.'s user avatar
-1 votes
1 answer
330 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 ...
Cat's user avatar
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