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\begin{document}

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\section*{supergroup}

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

\hypertarget{supergeometry}{}\paragraph*{{Supergeometry}}\label{supergeometry}

[[!include supergeometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{algebraic_super_groups}{Algebraic super groups}\dotfill \pageref*{algebraic_super_groups} \linebreak
\noindent\hyperlink{super_lie_groups}{Super Lie groups}\dotfill \pageref*{super_lie_groups} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_generalized_group_elements}{In terms of generalized group elements}\dotfill \pageref*{in_terms_of_generalized_group_elements} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{SuperTranslationGroup}{The super-translation group}\dotfill \pageref*{SuperTranslationGroup} \linebreak
\noindent\hyperlink{the_super_euclidean_group}{The super Euclidean group}\dotfill \pageref*{the_super_euclidean_group} \linebreak
\noindent\hyperlink{general_linear_supergroup}{General linear supergroup}\dotfill \pageref*{general_linear_supergroup} \linebreak
\noindent\hyperlink{orthosymplectic_supergroup}{Orthosymplectic supergroup}\dotfill \pageref*{orthosymplectic_supergroup} \linebreak
\noindent\hyperlink{finite_supergroups}{Finite super-groups}\dotfill \pageref*{finite_supergroups} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{representations_tannaka_duality_and_delignes_theorem}{Representations, Tannaka duality and Deligne's theorem}\dotfill \pageref*{representations_tannaka_duality_and_delignes_theorem} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{super-group} is the analog in [[supergeometry]] of [[Lie groups]] in [[differential geometry]].

\hypertarget{algebraic_super_groups}{}\subsection*{{Algebraic super groups}}\label{algebraic_super_groups}

An [[affine scheme|affine]] algebraic super group is the [[formal dual]] of a [[superalgebra|super]]-[[commutative Hopf algebra]].

\hypertarget{super_lie_groups}{}\subsection*{{Super Lie groups}}\label{super_lie_groups}

\hypertarget{definition}{}\subsubsection*{{Definition}}\label{definition}

A \emph{super Lie group} is a [[group object]] in the [[category]] [[SDiff]] of [[supermanifolds]], that is a super [[Lie group]].

\hypertarget{in_terms_of_generalized_group_elements}{}\paragraph*{{In terms of generalized group elements}}\label{in_terms_of_generalized_group_elements}

One useful way to characterize [[group object]]s $G$ in the [[category]] $SDiff$ of [[supermanifold]] is by first sending $G$ with the [[Yoneda embedding]] to a [[presheaf]] on $SDiff$ and then imposing a lift of $Y(G) : SDiff^{op} \to Set$ through the [[forgetful functor]] [[Grp]] $\to$ [[Set]] that sends a (ordinary) [[group]] to its underlying [[set]].

So a group object structure on $G$ is a diagram

\begin{displaymath}
\itexarray{
    && Grp
    \\
    & {}^{(G,\cdot)}\nearrow & \downarrow
    \\
    SDiff^{op} &\stackrel{Y(G)}{\to}& Set
  }
  \,.
\end{displaymath}
This gives for each [[supermanifold]] $S$ an ordinary group $(G(S), \cdot)$, so in particular a product operation

\begin{displaymath}
\cdot_S : G(S) \times G(S) \to G(S)
  \,.
\end{displaymath}
Moreover, since morphisms in $Grp$ are group homomorphisms, it follows that for every morphism $f : S \to T$ of [[supermanifold]]s we get a commuting diagram

\begin{displaymath}
\itexarray{
    G(S) \times G(S) &\stackrel{\cdot_S}{\to}& G(S)
    \\
    \uparrow^{G(f)\times G(f)}  && \uparrow^{G(f)}
    \\
    G(T) \times G(T) &\stackrel{\cdot_T}{\to}& G(T)     
  }
\end{displaymath}
Taken together this means that there is a morphism

\begin{displaymath}
Y(G \times G)  \to Y(G)
\end{displaymath}
of representable presheaves. By the [[Yoneda lemma]], this uniquely comes from a morphism $\cdot : G \times G \to G$, which is the product of the group structure on the object $G$ that we are after.

etc.

This way of thinking about supergroups is often explicit in some parts of the literature on supergeometry: some authors define a supergroup or [[super Lie algebra]] as a rule that assigns to every [[Grassmann algebra]] $A$ over an ordinary [[vector space]] an ordinary [[group]] $G(A)$ or [[Lie algebra]] and to a morphism of [[Grassmann algebra]]s $A \to B$ \emph{covariantly} a morphism of groups $G(A) \to G(B)$. But the [[Grassmann algebra]] on an $n$-dimensional [[vector space]] is naturally isomorphic to the function ring on the [[supermanifold]] $\mathbb{R}^{0|n }$. So the definition of supergroups in terms of Grassmann algebras is secretly the same as the above definition in terms of the [[Yoneda embedding]].

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

\hypertarget{SuperTranslationGroup}{}\paragraph*{{The super-translation group}}\label{SuperTranslationGroup}

also called the \textbf{[[super-Heisenberg group]]}

The additive group structure on $\mathbb{R}^{1|1}$ is given on [[generalized element]]s in (i.e. in the logic internal to) the [[topos]] of [[sheaf|sheaves]] on the category [[SCartSp]] of [[cartesian space|cartesian]] superspaces by

\begin{displaymath}
\mathbb{R}^{1|1} \times \mathbb{R}^{1|1} \to 
  \mathbb{R}^{1|1}
\end{displaymath}
\begin{displaymath}
(t_1, \theta_1), (t_2, \theta_2)
  \mapsto
  (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2)
  \,.
\end{displaymath}
Recall how the notation works here: by the [[Yoneda embedding]] we have a [[full and faithful functor]]

\begin{quote}%
[[SDiff]] $\hookrightarrow$ $Fun(SDiff^{op}, Set)$


\end{quote}
and we also have the theorem, discussed at [[supermanifold]]s, that maps from some $S \in SDiff$ into $\mathbb{R}^{p|q}$ is given by a tuple of $p$ even section $t_i$ and $q$ odd sections $\theta_j$. The above notation specifies the map of supermanifolds by displaying what map of sets of maps from some test object $S$ it corresponds to under the [[Yoneda embedding]].

Now, for each $S \in$ [[SDiff]] there is a [[group]] structure on the [[hom-set]] $SDiff(S, \mathbb{R}^{1|1}) \simeq C^\infty(S)^{ev} \times C^\infty(S)^{odd}$ given by precisely the above formula for this given $S$

\begin{displaymath}
\mathbb{R}^{1|1}(S) \times \mathbb{R}^{1|1}(S) \to 
  \mathbb{R}^{1|1}(S)
\end{displaymath}
\begin{displaymath}
(t_1, \theta_1), (t_2, \theta_2)
  \mapsto
  (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2)
  \,.
\end{displaymath}
where $(t_i, \theta_i) \in C^\infty(S)^{ev} \times C^\infty(S)^{odd}$ etc and where the addition and product on the right takes place in the function [[super algebra]] $C^\infty(S)$.

Since the formula looks the same for all $S$, one often just writes it without mentioning $S$ as above.

\hypertarget{the_super_euclidean_group}{}\paragraph*{{The super Euclidean group}}\label{the_super_euclidean_group}

The super-translation group is the $(1|1)$-dimensional case of the [[super Euclidean group]].

\hypertarget{general_linear_supergroup}{}\paragraph*{{General linear supergroup}}\label{general_linear_supergroup}

[[general linear supergroup]]

\ldots{}

\hypertarget{orthosymplectic_supergroup}{}\paragraph*{{Orthosymplectic supergroup}}\label{orthosymplectic_supergroup}

[[orthosymplectic supergroup]]

\ldots{}

\hypertarget{finite_supergroups}{}\subsection*{{Finite super-groups}}\label{finite_supergroups}

There is a finite analog for super-groups that does not quite fit in the framework presented here:

\begin{defn}
\label{}\hypertarget{}{}
A finite super-group is a tuple $(G, z \in G)$, where $G$ is a finite group and $z$ is central and squares to $1$.

The representations of a finite super-group are $\mathbb{Z}_2$-graded: An irreducible representation has odd degree if $z$ acts by negation, and even degree if it acts as the identity.

This definition is found e.g. in:

\begin{itemize}%
\item Paul Bruillard, Cesar Galindo, Tobias Hagge, Siu-Hung Ng, Julia Yael Plavnik, Eric C. Rowell, Zhenghan Wang, \emph{Fermionic Modular Categories and the 16-fold Way} (\href{https://arxiv.org/pdf/1603.09294v2}{pdf})

\end{itemize}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{representations_tannaka_duality_and_delignes_theorem}{}\subsubsection*{{Representations, Tannaka duality and Deligne's theorem}}\label{representations_tannaka_duality_and_delignes_theorem}

[[Deligne's theorem on tensor categories]] (see there for details) says that every suitably well-behave linear [[tensor category]] is the [[category of representations]] of an algebraic supergroup. In particular the [[Hopf algebra]] of functions on an affine algebraic supergroup is a [[triangular Hopf algebra]].

[[!include structure on algebras and their module categories - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Dennis Westra]], \emph{Superrings and supergroups}, 2009 (\href{http://www.mat.univie.ac.at/~michor/westra_diss.pdf}{pdf})


\item Groeger, \emph{Super Lie groups and super Lie algebras}, lecture notes 2011 (\href{http://www.mathematik.hu-berlin.de/~groegerj/teaching_files/lecture12.pdf}{pdf})


\item [[Veeravalli Varadarajan]], section 7.1 of \emph{[[Supersymmetry for mathematicians]]: An introduction}



\end{itemize}
Discussion of [[group extensions]] of supergroups includes

\begin{itemize}%
\item [[Christopher Drupieski]], \emph{Strict polynomial superfunctors and universal extension classes for algebraic supergroups} (\href{http://arxiv.org/abs/1408.5764}{arXiv:1408.5764})

\end{itemize}
Discussion as Hopf-superalgebras includes

\begin{itemize}%
\item Nicol\'a{}s Andruskiewitsch, Iv\'a{}n Angiono, Hiroyuki Yamane, \emph{On pointed Hopf superalgebras}, Contemp. Math. vol. 544, pp. 123--140, 2011 (\href{https://arxiv.org/abs/1009.5148}{arXiv:1009.5148})

\end{itemize}
[[!redirects supergroups]]

[[!redirects super Lie group]] [[!redirects super Lie groups]]

[[!redirects super group]] [[!redirects super groups]]

[[!redirects Lie supergroup]] [[!redirects Lie supergroups]]

[[!redirects super-group]] [[!redirects super-groups]]



\end{document}