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\begin{document}

%-------------------------------------------------------------------


\section*{super Euclidean group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{}{$d = 1$}\dotfill \pageref*{} \linebreak
\noindent\hyperlink{_2}{$d = 2$}\dotfill \pageref*{_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The ordinary [[Euclidean group]] of $\mathbb{R}^n$ is the group generated from the rigid translation action of $\mathbb{R}^n$ on itself and rotations about the origin.

The \emph{super Euclidean group} is analogously the [[supergroup]] of translations and rotations of the [[supermanifold]] $\mathbb{R}^{p|q}$.

Its [[super Lie algebra]] should be the [[super Poincare Lie algebra]] (up to the signature of the metric).

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\begin{quote}%
\textbf{incomplete} for the moment, to be finished off tomorrow


\end{quote}
The following description of the super Euclidean group (once it is finished, and polished) is due to [[Stephan Stolz]] and [[Peter Teichner]].

The data needed to define the super Euclidean group is

\begin{enumerate}%
\item $V$ a $d$-dimensional inner product space


\item a [[spinor representation]] $\Delta^*$ of $Spin(V)$


\item a $Spin(V)$-equivariant map

\begin{displaymath}
\Gamma : \Delta^* \otimes_{\mathbb{C}}
    \Delta^* \to V \otimes_{\mathbb{R}} \mathbb{C}
\end{displaymath}


\end{enumerate}
where $Spin(V)$ is the [[Spin group]] (see [[Clifford algebra]] for the moment).

Here is the construction of $Eucl(\mathbb{R}^{d|\delta})$ for

\begin{itemize}%
\item $d = dim_{\mathbb{R}} V$

\end{itemize}
\textbf{remark} $\delta$ is a multiple of $2^{[\frac{d-1}{2}]}$

set

\begin{displaymath}
X = \Pi
  \left(
    \itexarray{ V \times \Delta^* \\ \downarrow \\ V}
  \right)
  = 
  V \times \Pi \Delta^*
\end{displaymath}
is a [[complex supermanifold]] of dimension $(d|\delta)$

\begin{displaymath}
C^\infty(V \times \Pi \Delta^*)
  =
  C^\infty(V) \otimes 
  \wedge^\bullet \Delta
  = 
  C^\infty(V, \wedge^\bullet(\Delta))
\end{displaymath}
for $\delta = 1$ this is

\begin{displaymath}
\cdots = C^\infty(C, \wedge^\bullet \Delta)^{ev}
    \oplus
   C^\infty(C, \wedge^\bullet \Delta)^{odd}
\end{displaymath}
\begin{displaymath}
\cdots \simeq
    C^\infty(V) \oplus C^\infty(V, \wedge^\bullet \Delta)
\end{displaymath}
where the last factor is $\simeq C^\infty(V; \Delta) \simeq C^\infty(V, S^+)$ where $S^+$ is the [[spinor bundle]]

now define the multiplication

\begin{displaymath}
(V \times \Pi \Delta^*)
  \times
  (V \times \Pi \Delta^*)
  \stackrel{\mu}{\to}
  (V \times \Pi \Delta^*)
\end{displaymath}
by sayin what it does on sets of probes by $S$

\begin{displaymath}
(V \times \Pi \Delta^*)(S)
  \times
  (V \times \Pi \Delta^*)(S)
  \stackrel{\mu(S)}{\to}
  (V \times \Pi \Delta^*)(S)
\end{displaymath}
here on the left we have the sets of sections

\begin{displaymath}
C^\infty(S)^{ev} \otimes V
  \times
  C^\infty(S)^{ev} \otimes \Delta^*
\end{displaymath}
so we can map these as

\begin{displaymath}
((v_1, \theta_1),
   (v_2, \theta_2)
  )
  \mapsto
  (v_1 + v_2 + \Gamma(\theta_1 \otimes \theta_2), 
   \theta_1 + \theta_2)
\end{displaymath}
\textbf{Remark}

if the data $(V, \Delta^*, \Gamma)$ and $(V', (\Delta^*)', \Gamma')$ is [[isomorphism|isomorphic]] we get compatible notions of structures

But if $d = 0,1,2$ and $\delta = 1$ then there is a \emph{unique} such triple with non-degenerate pairing $\Gamma$ up to isomorphism.

\textbf{Definition}

The structure of a \textbf{[[Euclidean supermanifold]]} on a $(d|\delta)$-dimensional [[supermanifold]] $Y$ is a $(V \times \Pi \Delta^*, End(V, \Delta^*, \Gamma))$-structure. See there for details.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

recall the [[Clifford algebra]] table:

\begin{displaymath}
\itexarray{
      d & Cl(\mathbb{R}^d)^{ev} & Spin(\mathbb{R}^d)
      \\
      \\
      1 & \mathbb{R} & \{\pm 1\}
      \\
      2 & (\mathbb{R} 1 \oplus \mathbb{R} e_1 e_2, e_i^2 = 1) \simeq \mathbb{C} & S^1
  }
\end{displaymath}
the group structure on $V \times \Pi \Delta^*$ is that of the ``translations'' and ``rotations''

it will be defined on [[generalized element]]s with domain $S$ by maps of sets

\begin{displaymath}
\mu:
  (V \times \Pi \Delta^*)(S)
  \times
  (V \times \Pi \Delta^*)(S)
  \to
  (V \times \Pi \Delta^*)(S)
\end{displaymath}
\begin{displaymath}
(v_1, \theta_1) , (v_2, \theta_2)
  \mapsto
  (v_1+ v_2 + \Gamma(\theta_1 \otimes \theta_2),
  \theta_1 + \theta_2)
\end{displaymath}
\hypertarget{}{}\subsection*{{$d = 1$}}\label{}

$\Delta^* = \mathbb{C}$

\begin{displaymath}
\Gamma : \Delta^* \otimes \Delta^* \to V \otimes_{\mathbb{R}} \mathbb{C}
\end{displaymath}
\begin{displaymath}
\mathbb{C} \otimes \mathbb{C}
  \to
  \mathbb{R} \otimes_{\mathbb{R}} \mathbb{C}
\end{displaymath}
\begin{displaymath}
1 \otimes 1 \mapsto 1 \otimes 1
\end{displaymath}
so here this is the [[supergroup|super translation group]].

\hypertarget{_2}{}\subsection*{{$d = 2$}}\label{_2}

$\Delta^* = \mathbb{C}$

\begin{displaymath}
u \in S^1 \simeq U(1) \simeq Spin(\mathbb{R}^2)
\end{displaymath}
\begin{displaymath}
\Delta^* \otimes_{\mathbb{C}}
  \Delta^*
  \stackrel{\Gamma}{\to}
  \mathbb{R}^2 \otimes \mathbb{R}
  \simeq \mathbb{C} \oplus \mathbb{C}
\end{displaymath}
the first map is multiplication by $u^{-1}$ and then the isomorphism on the right sends

\begin{displaymath}
(x,y)\otimes 1 \mapsto (z, \bar z)
\end{displaymath}
where $z = x + i y$

translation group $V \times \Pi \Delta^* \simeq \mathbb{R}^{2|1}$

multiplication on $S$-elements

\begin{displaymath}
\mathbb{R}^{2|1}(S) \times
  \mathbb{R}^{2|1}(S)
  \to 
  \mathbb{R}^{2|1}(S)
\end{displaymath}
given by

\begin{displaymath}
(z_1,\bar z_1, \theta_1),
  (z_2,\bar z_2, \theta_2)
  \mapsto
  (z_1 + z_2, \bar z_1 + \bar z_2 + \theta_1 \theta_2, \theta_1 + \theta_2)
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Euclidean group]]


\item [[general linear supergroup]], [[orthosymplectic supergroup]]


\item [[Lorentz group]], [[Poincare group]]


\item [[super Poincaré group ]]



\end{itemize}
[[!redirects Euclidean supergroup]]



\end{document}