A Super Version of the Connes-Moscovici Hopf Algebra

TL;DR

This paper introduces a super version of the Connes-Moscovici Hopf algebra by constructing a bicrossproduct super Hopf algebra based on supergroups of diffeomorphisms and affine transformations of the superline.

Contribution

It defines a new super Hopf algebra $\\mathcal{H}_1^s$ as a bicrossproduct of supergroup Hopf algebras, extending the classical Connes-Moscovici algebra to the super setting.

Findings

Explicit description of the super Hopf algebra $\\mathcal{H}_1^s$ in terms of generators and relations.

Construction of the supergroup $G^s$ and its subgroups $G^s_1$, $G^s_2$.

Extension of the classical Connes-Moscovici algebra to a supergeometric context.

Abstract

We define a super version of the Connes-Moscovici Hopf algebra, $\mathcal{H}_1$. For that, we consider the supergroup $G^s=Diff^+(\mathbb{R}^{1,1})$ of orientation preserving diffeomorphisms of the superline $\mathbb{R}^{1,1}$ and define two (super) subgroups $G^s_1$ and $G^s_2$ of $G^s$ where $G^s_1$ is the supergroup of affine transformations. The super Hopf algebra $\mathcal{H} ^s_1$ is defined as a certain bicrossproduct super Hopf algebra of the super Hopf algebras attached to $G^s_1$ and $G^s_2$. We also give an explicit description of $\mathcal{H} ^s_1$ in terms of generators and relations.