The graph complex acts on the spaces of Poisson bi-vectors \(\mathcal{P}\) by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. \(\mathcal{P} = {{{\text{L}}}_{{\vec {V}}}}(\mathcal{P})\) w.r.t. the Lie derivative along some vector field \(\vec {V}\), but not quadratic (the coefficients of \(\mathcal{P}\) are not degree-two homogeneous polynomials), and whenever its velocity bi-vector \(\dot {\mathcal{P}} = \mathcal{Q}(\mathcal{P})\), also homogeneous w.r.t. \(\vec {V}\) by \({{{\text{L}}}_{{\vec {V}}}}(\mathcal{Q}) = n\mathcal{Q}\) whenever \( is obtained using the orientation morphism \({\text{O}\vec {r}}\) from a graph cocycle \(\gamma \) on \(n\) vertices and \(2n - 2\) edges, then the \(1\)-vector \(\vec {\mathcal{X}} = {\text{O}\vec {r}}(\gamma )(\vec {V} \otimes {{\mathcal{P}}^{{{{ \otimes }^{{n - 1}}}}}})\) is a Poisson cocycle. Its construction is uniform for all Poisson bi-vectors \(\mathcal{P}\) satisfying the above assumptions, on all finite-dimensional affine manifolds \(M\). Still, if the bi-vector \(\mathcal{Q}{\not \equiv }0\) is exact in the respective Poisson cohomology, so there exists a vector field \(\vec {\mathcal{Y}}\) such that \(\mathcal{Q}(\mathcal{P}) = \left[\kern-0.15em\left[ {\vec {\mathcal{Y}},\mathcal{P}} \right]\kern-0.15em\right]\), then the universal cocycle \(\vec {\mathcal{X}}\) does not belong to the coset of \(\vec {\mathcal{Y}}\) mod \(\text{ker}\left[\kern-0.15em\left[ {\mathcal{P}, \cdot } \right]\kern-0.15em\right]\). We illustrate the construction using two examples of cubic-coefficient Poisson brackets associated with the \(R\)‑matrices for the Lie algebra \(\mathfrak{g}\mathfrak{l}(2)\).

中文翻译：

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通用Cocycles和图复杂性对同构Poisson括号的影响

图复数通过无穷小对称性作用于泊松双向量\（\ mathcal {P} \）的空间上。我们证明只要Poisson结构是齐构的，即\（\ mathcal {P} = {{{\ text {L}}} _ {{\ vec {V}}}}（\ mathcal {P}）\）wrt沿某些矢量场\（\ vec {V} \）的Lie导数，但不是二次方的（\（\ mathcal {P} \）的系数不是二阶齐次多项式），并且每当它的速度双矢量\ （\ dot {\ mathcal {P}} = \ mathcal {Q}（\ mathcal {P}）\），也由\（{{{\ text {L}} } _ {{\ vec {V}}}}（\ mathcal {Q}）= n \ mathcal {Q} \）每当使用定向态射\\ {{\ text {O} \ vec {r} } \）来自\（n \）顶点和\（2n-2 \）边上的图循环\（\ gamma \），那么\（1 \）-向量\（\ vec {\ mathcal {X}} = {\ text {O} \ vec {r}}（\ gamma）（\ vec {V} \ otimes {{\ mathcal { P}} ^ {{{{\ otimes} ^ {{n-1}}}}}}）））是泊松Cocycle。对于所有满足上述假设的Poisson双矢量\（\ mathcal {P} \），在所有有限维仿射流形\（M \）上，其构造都是统一的。不过，如果双向量\（\ mathcal {Q} {\ not \ equiv} 0 \）在相应的Poisson谐函数中是精确的，则存在一个向量场\（\ vec {\ mathcal {Y}} \\这样\（\ mathcal {Q}（\ mathcal {P}）= \ left [\ kern-0.15em \ left [{\ vec {\ mathcal {Y}}，\ mathcal {P}} \ right] \ kern -0.15em \ right] \），则通用循环\（\ vec {\ mathcal {X}} \）不属于\（\ vec {\ mathcal {Y}} \} mod \（\文字{ker} \ left [\ kern-0.15em \ left [{\ mathcal {P}，\ cdot} \ right] \ kern-0.15em \ right] \）。