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

Agrebaoui, Boujemaa ; Hattab, Raja

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1143-1163.

Résumé

The relative cohomology ${\mathrm{H}}_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^1(\mathbb{K}(1|3),\mathfrak{osp}(2,3);{\mathcal{D}}_{\lambda ,\mu }(S^{1|3}))$ 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 $s(X_f)=D_1{D}_2{D}_3(f){\alpha }_3^{1/2}$, $X_f\in \mathbb{K}(1|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathfrak{osp}(2,3)$ of $\mathbb{K}(1|3)$. \endgraf 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 ${\mathrm{H}}_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^1(\mathcal{K}(1|3)$, $\mathrm{P}\mathrm{C}(2,3)$; ${\mathcal{D}}_{\lambda ,\mu }(S^{1|3})$ 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 $S_3(\mathrm{\Phi })=E_{\mathrm{\Phi }}^{-1}(D_1(D_2),D_3){\alpha }_3^{1/2}$, $\mathrm{\Phi }\in \mathcal{K}(1|3)$ introduced by Radul which is invariant with respect to the conformal group $\mathrm{P}\mathrm{C}(2,3)$ of $\mathcal{K}(1|3)$. These cocycles are expressed in terms of $S_3(\mathrm{\Phi })$ and possess its properties.

Détail

DOI : 10.1007/s10587-016-0315-5

Classification : 13N10, 17B56, 17B66, 20G10, 20J06, 53D10, 58A50
Mots-clés : contact vector field; cohomology of groups; group of contactomorphisms; super-Schwarzian derivative; invariant differential operator