发表机构
Université de Gafsa, Faculté des Sciences; Université de Sfax, Faculté des Sciences（加夫萨大学理学院; 萨法克斯大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究$\frak{osp}(n|2)$在$\reals^{1|n}$微分算子符号超空间上作用的形式形变，计算二次阻碍，给出可积性条件及通用形变性质，证明可积形变等价于其无穷小部分。
AI 中文摘要
我们研究$\boldsymbol{n\bold{\textgreater} 3}$时，$\boldsymbol{\reals^{1|n}}$上加权密度的线性微分算子符号超空间$\boldsymbol{\frak{S}^n_d=\bigoplus_{k\bold{\textgreater} 0}\frak{F}^n_{d-\frac{k}{2}}}$上自然$\boldsymbol{\frak{osp}(n|2)}$作用的形式形变。从文献\textbf{[10]}中计算得到的一阶上同调空间出发，我们计算携带二次阻碍的上积$\boldsymbol{\text{H}^1\bold{\text{∨}} \text{H}^1\to \text{H}^2}$。该结果由$\boldsymbol{\frak{osp}(n|2)}$不变算子$\boldsymbol{A_k=\boldsymbol{\text{η}_1\boldsymbol{\text{⋯}}\text{η}_n\boldsymbol{\text{∂}}_x^{k-1}}}$决定：张成$\text{H}^1$非对角部分的两个上同调类$\boldsymbol{h_k}$和$\boldsymbol{\tilde{h}_k}$恰好是$\boldsymbol{A_k}$的上边缘对两个权重的两个导数。因此，所有两个非对角类的乘积以及所有两个对角类的乘积均消失，对于每个$\boldsymbol{k}$，整个阻碍由单个非平凡二阶上同调类$\boldsymbol{\text{Ω}_k}$承担。若$\boldsymbol{2d\notin\reals}$，则空间$\boldsymbol{\text{H}^1\bold{\text{∨}}\text{H}^1}$恒为零，故每个无穷小形变均可积；若$\boldsymbol{2d=m\bold{\text{∈}}\reals}$，则得到恰好$\boldsymbol{m}$个二次可积性条件$\boldsymbol{\tau_{2-n-k}(t_k-\tilde{t}_k)+\tau_k\tilde{t}_k=0}$（$\boldsymbol{1\bold{\text{≤}}k\bold{\text{≤}}m}$），我们证明这些条件也是充分的：不存在阶$\boldsymbol{\bold{\text{≥}}3}$的条件，且通用形变在参数中为一次度。特别地，每个可积形式形变都等价于其无穷小部分。
英文摘要
We study formal deformations of the natural $\osp(n|2)$-action, $n\geq 3$, on the superspace $\Sc^n_d=\bigoplus_{k\geq 0}\Fc^n_{d-\frac{k}{2}}$ of symbols of linear differential operators on weighted densities over $\R^{1|n}$. Starting from the first cohomology space computed in \cite{10}, we compute the cup-product $\Hd^1\vee \Hd^1\to \Hd^2$ which carries the quadratic obstructions. The answer is governed by the $\osp(n|2)$-invariant operators $A_k=η_1\cdotsη_n\partial_x^{k-1}$: the two cocycles $h_k$ and $\widetilde{h}_k$ spanning the off-diagonal part of $\Hd^1$ are exactly the two derivatives of the coboundary of $A_k$ with respect to the two weights. Consequently, all the products of two off-diagonal classes and all the products of two diagonal classes vanish, and the whole obstruction is carried, for each $k$, by a single non-trivial 2-cocycle $Ω_k$. If $2d\notin\N$ the space $\Hd^1\vee\Hd^1$ is identically zero, so every infinitesimal deformation is integrable. If $2d=m\in\N$ we obtain exactly $m$ quadratic integrability conditions, $τ_{2-n-k}(t_k-\widetilde{t}_k)+τ_k\widetilde{t}_k=0$, $1\leq k\leq m$, and we prove that they are also sufficient: no condition of order $\geq 3$ occurs and the versal deformation is of degree one in the parameters. In particular every integrable formal deformation is equivalent to its infinitesimal part.
Comments14 pages
