$1$-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 ${\rm H}^1_{\rm diff}(\mathbb {K}(1|3),\mathfrak {osp}(2,3);{\mathcal {D}}_{\lambda ,\mu }(S^{1|3}))$ 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 $\mathbb {K}(1|3)$. \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 ${\rm H}^1_{\rm diff}({\mathcal {K}}(1|3)$, ${\rm PC}(2,3)$; ${\mathcal {D}}_{{\lambda },\mu }(S^{1|3})$ 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 ${\rm PC}(2,3)$ of ${\mathcal {K}}(1|3)$. These cocycles are expressed in terms of $S_{3}(\Phi )$ and possess its properties.

Détail

DOI : 10.1007/s10587-016-0315-5

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