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.
127
questions
2
votes
0
answers
32
views
Reconstruction of a Poisson-Lie group structure from a Lie bialgebra $\mathfrak{g}$
Let $(\mathfrak{g}, [,], \delta)$ be a Lie bialgebra where $\delta$ is the cobracket. It is well-known that there exists a simply connected Poisson-Lie group $G$ such that $\mathfrak{g} = \mathrm{Lie}(...
- 315
4
votes
1
answer
123
views
The Poisson center of a symplectic variety
Let $X$ be a symplectic (affine) variety over $\mathbb{C}$, that is, a normal variety with a non-degenerate closed (algebraic) 2-form on the smooth locus. How can we deduce that the Poisson center is ...
- 731
1
vote
0
answers
59
views
Poisson structure which is non-degenerate on the smooth locus
I hope this is not a silly question. I recently started studying Poisson structures on algebraic varieties.
Here is my question: Let $X$ be an algebraic variety, possibly not smooth. If $X$ is even ...
- 147
0
votes
0
answers
64
views
Some details about Kirillov-Kostant Poisson bracket
Let $G$ be a finite dimensional Lie group with Lie algebra $\mathfrak{g}$. The Kirillov-Kostant Poisson bracket on $\mathfrak{g}^*$ is defined as $$\{\cdot ,\cdot \} :C^{\infty}(\mathfrak{g}^*)\times ...
- 287
2
votes
1
answer
174
views
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 ...
- 523
2
votes
0
answers
179
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, ...
- 832
1
vote
0
answers
67
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 \...
- 829
3
votes
0
answers
516
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\...
- 442
0
votes
0
answers
100
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 ...
- 315
2
votes
0
answers
181
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 ...
- 5,993
0
votes
0
answers
138
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
28
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 ...
- 599
0
votes
0
answers
69
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$...
- 199
0
votes
0
answers
68
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}}) \...
- 199
1
vote
0
answers
40
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_{...
- 199
1
vote
0
answers
64
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 $\...
- 199
2
votes
1
answer
173
views
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 ...
- 5,993
4
votes
1
answer
374
views
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 ...
- 153
5
votes
0
answers
228
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 ...
- 494
3
votes
0
answers
228
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 ...
- 137
6
votes
1
answer
378
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$...
- 819
1
vote
2
answers
320
views
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 ...
- 5,993
5
votes
0
answers
121
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 $(...
- 181
3
votes
0
answers
92
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 (...
- 1,703
3
votes
1
answer
141
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 "...
- 1,154
1
vote
0
answers
81
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 ...
- 181
2
votes
1
answer
359
views
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 ...
- 21
3
votes
0
answers
126
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 ...
- 31
4
votes
0
answers
122
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 $...
- 661
1
vote
1
answer
272
views
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 ...
- 1,154
2
votes
1
answer
124
views
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?
- 1,154
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. ...
- 55
3
votes
0
answers
83
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 ...
user108998
6
votes
0
answers
211
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 ...
- 517
1
vote
0
answers
64
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$ ...
- 774
2
votes
1
answer
539
views
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 ...
- 175
3
votes
3
answers
264
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, ...
- 359
4
votes
0
answers
155
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?
- 3,870
3
votes
0
answers
85
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 $\...
user108998
4
votes
0
answers
125
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 ...
- 241
1
vote
1
answer
202
views
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 ...
- 323
3
votes
0
answers
250
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, ...
- 31
3
votes
0
answers
202
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 ...
- 327
6
votes
1
answer
671
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 ...
- 4,991
4
votes
0
answers
218
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 ...
user108998
3
votes
1
answer
271
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 ...
- 969
3
votes
0
answers
269
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 ...
- 4,975
11
votes
2
answers
560
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 ...
- 6,297
2
votes
1
answer
327
views
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 ...
- 23
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 ...