$1$-cocycles on the group of contactomorphisms on the supercircle $S^{1|3}$ generalizing the Schwarzian derivative

Czechoslovak Mathematical Journal

Boujemaa Agrebaoui¹  Raja Hattab² 

[1] Faculty of Sciences, Sfax University, P.B.: 1171, Soukra Road KM 3.5, Avenue Salaheddine Ayoubi, 3000 Sfax, Tunisia;Higher Institute Systems Industrial De Gabés, Gabés University, Avenue Salaheddine El Ayoubi, 6000 Gabes, Tunisia

关键词: contact vector field;    cohomology of groups;    group of contactomorphisms;    super-Schwarzian derivative;    invariant differential operator;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

The relative cohomology $ H^1_ diff(\mathbb{K}(1|3),\mathfrak{osp}(2,3);{\mathcal{D}}_{\lambda,\mu}(S^{1|3}))$ of the contact Lie superalgebra $\mathbb{K}(1|3)$ with coefficients in the space of differential operators ${\mathcal{D}}_{\lambda,\mu}(S^{1|3})$ acting on tensor densities on $S^{1|3}$, is calculated in {N. Ben Fraj, I. Laraied, S. Omri} (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s(X_f)=D_1D_2D_3(f)\alpha_3^{1/2}$, $X_f\in\mathbb{K}(1|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathfrak{osp}(2,3)$ of $\mathbb{K}(1|3)$.In this work we study the supergroup case. We give an explicit construction of $1$-cocycles of the group of contactomorphisms ${\mathcal{K}}(1|3)$ on the supercircle $S^{1|3}$ generating the relative cohomology $ H^1_ diff({\mathcal{K}}(1|3)$, $ PC(2,3)$; ${\mathcal{D}}_{{\lambda},\mu}(S^{1|3})$ with coefficients in ${\mathcal{D}}_{{\lambda},\mu}(S^{1|3})$. We show that they possess properties similar to those of the super-Schwarzian derivative $1$-cocycle $S_3(\Phi)=E_{\Phi}^{-1}(D_1(D_2),D_3)\alpha_3^{1/2}$, $\Phi\in{\mathcal{K}}(1|3)$ introduced by Radul which is invariant with respect to the conformal group $ PC(2,3)$ of ${\mathcal{K}}(1|3)$. These cocycles are expressed in terms of $S_3(\Phi)$ and possess its properties.

【 授权许可】

Unknown