Tagged Questions

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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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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Super classical r-matrices and Poisson Lie supergroups

In classical case, given an r-matrix $r$ for $sl_n$, we can compute the corresponding Poisson bracket on $SL_n$ by using the formula $\{L \otimes L\} = [r, L \otimes L]$. For example, let $g=sl_2$ and ...
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Closing the commutative diagram for symplectic realizations

Let $f: (M_1, P_1) \to (M_2, P_2)$ be a Poisson map between Poisson manifolds. Let $\pi_i : (S_i, \omega_i) \to (M_i, P_i), \ i=1,2$ be symplectic realizations. Putting these objects in a rectangular ...
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Recover Poisson bracket on $C[G]$ using the Lie cobracket $\delta: g \to \Lambda^2 g$

By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on ...
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Poisson ideals vs. ideals generated by Poisson central elements

Let $R$ be a Poisson algebra (over $\mathbb C$, say) with Poisson center $Z = \{c \in R : \{c,R\} = 0\}$ and consider two types of ideals $I \leq R$: $I = \langle (c_i) \rangle$ is generated by ...
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Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space?

It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more ...
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What is twisted in a twisted Poisson structure?

The usual definition of a twisted Poisson algebra involving a 3-form does not seem to refer to a Poisson algebra that is then twisted, i.e. twisting either the commutative product or the Lie bracket. ...
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Absent 2nd order terms in deformation quantization of Poisson manifolds

I am reading Kontsevich' famous paper on deformation quantization of Poisson manifolds. In section 1.4.2 on page 4 he gives the general formula for the star product associated to a Poisson structure ...
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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 ...
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Which commutative algebras admit a nonzero Poisson bracket?

Let $A$ be a commutative algebra, not necessarily unital, over a field $k$ (of characteristic not equal to $2$, or even equal to $0$, if it helps). A second-order formal deformation of $A$ is a \$k[h]/...
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Quantization and noncommutative deformations

Hello, I would like to introduce myself to the theory of quantization and noncommutative deformations of Riemann Poisson structures. In fact, I am familiar with Riemannian and Poisson geometry, but I ...
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Courant algebroids from Poisson geometry

Hello! Could somebody provide some simple examples of Courant algebroids coming from Poisson geometry? Thanks!