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.
48
questions with no upvoted or accepted answers
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
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
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
111
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
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
0
answers
509
views
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\...
3
votes
0
answers
222
views
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 ...
3
votes
0
answers
88
views
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 (...
3
votes
0
answers
124
views
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 ...
3
votes
0
answers
126
views
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. ...
3
votes
0
answers
82
views
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 ...
3
votes
0
answers
84
views
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 $\...
3
votes
0
answers
243
views
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, ...
3
votes
0
answers
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 ...
3
votes
0
answers
146
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 ^{...
3
votes
0
answers
267
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 ...
3
votes
0
answers
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 ...
2
votes
0
answers
166
views
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, ...
2
votes
0
answers
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 ...
2
votes
0
answers
181
views
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 ...
2
votes
0
answers
41
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 ...
2
votes
0
answers
216
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 ...
2
votes
0
answers
114
views
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 $...
1
vote
0
answers
56
views
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 \...
1
vote
0
answers
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 ...
1
vote
0
answers
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 ...
1
vote
0
answers
38
views
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_{...
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 $\...
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 ...
1
vote
0
answers
60
views
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$ ...
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 ...
1
vote
0
answers
59
views
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 ...
1
vote
0
answers
67
views
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 ...
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 ...
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 ...
0
votes
0
answers
88
views
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 ...
0
votes
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$...
0
votes
0
answers
64
views
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}}) \...
-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 ...