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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Poisson quantization vs quantization in atomic physics
Is it possible to interpret quantization in atomic physics ( e.g. the quantization condition in hydrogen atom stated as exponential decay of wave functions at infinity and analogously for n-electron ...
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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, ...
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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 \...
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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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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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. $r$...
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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 $$\pi^R (x) = (d_x R_{x^{-1}} \otimes d_x R_{x^{-1}}) \...
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Differential of tensor product of maps
Let $G$ be a Poisson-Lie group with Poisson bivector field $\pi.$ Let $\pi^{R}$ be the trivialization of $\pi$ with respect to the right translations i.e. $$\pi^{R} (g) = (d_{g} R_{g^{-1}} \otimes d_{...
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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 $\...
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Computations of certain Poisson cohomology groups
I am reading the paper Grothendieck groups of Poisson vector bundles by Viktor L. Ginzburg.
In that paper, the author introduces a new invariant for Poisson manifolds; which is called as the Poisson ...
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References on Namikawa-Weyl group
What are the most reasonable references on the definition of the Namikawa-Weyl groups and the first results about them?
In particular, are there more recent (or more educational) texts than the ...
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Is there a name for a noncommutative generalization of Poisson algebra?
Is there a name for an associative algebra which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e.,
$$[a, b \circ c]=[a,b]\circ c+b\circ [a,c], \hspace{15mm}(*)$$ ...
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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 ...
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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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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 ...
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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 $-2$...
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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 ...
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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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Linear poisson structures on vector bundles
A Poisson structure on a smooth manifold $M$ is a map $C^\infty(M)\times C^\infty(M)\times C^\infty(M)$ satisfying certain conditions.
For a vector space $V$, we can talk about a Poisson structure on ...
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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 $(...
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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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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 "...
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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 (...
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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 ...
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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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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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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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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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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 $...
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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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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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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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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 $\...
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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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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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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 ...