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

Agrebaoui, Boujemaa, and Hattab, Raja. "$1$-cocycles on the group of contactomorphisms on the supercircle $S^{1|3}$ generalizing the Schwarzian derivative." Czechoslovak Mathematical Journal 66.4 (2016): 1143-1163. <http://eudml.org/doc/287534>.

@article{Agrebaoui2016,
abstract = {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 $\mathbb \{K\}(1|3)$. 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.},
author = {Agrebaoui, Boujemaa, Hattab, Raja},
journal = {Czechoslovak Mathematical Journal},
keywords = {contact vector field; cohomology of groups; group of contactomorphisms; super-Schwarzian derivative; invariant differential operator; contact vector field; cohomology of groups; group of contactomorphisms; super-Schwarzian derivative; invariant differential operator},
language = {eng},
number = {4},
pages = {1143-1163},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$1$-cocycles on the group of contactomorphisms on the supercircle $S^\{1|3\}$ generalizing the Schwarzian derivative},
url = {http://eudml.org/doc/287534},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Agrebaoui, Boujemaa
AU - Hattab, Raja
TI - $1$-cocycles on the group of contactomorphisms on the supercircle $S^{1|3}$ generalizing the Schwarzian derivative
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 4
SP - 1143
EP - 1163
AB - 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 $\mathbb {K}(1|3)$. 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.
LA - eng
KW - contact vector field; cohomology of groups; group of contactomorphisms; super-Schwarzian derivative; invariant differential operator; contact vector field; cohomology of groups; group of contactomorphisms; super-Schwarzian derivative; invariant differential operator
UR - http://eudml.org/doc/287534
ER -