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\section*{orthosymplectic super Lie algebra}

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

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

[[!include supergeometry - contents]]

\hypertarget{lie_theory}{}\paragraph*{{Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_superstring_theory}{In superstring theory}\dotfill \pageref*{in_superstring_theory} \linebreak
\noindent\hyperlink{representation_theory}{Representation theory}\dotfill \pageref*{representation_theory} \linebreak


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

The \emph{orthosymplectic supergroup} $OSp(N|2p)$ is the sub-[[supergroup]] of the [[general linear supergroup]] $GL(N|2p)$ on those elements which preserve the canonical graded-symmetric [[bilinear form]] on $\mathbb{R}^{N|2p}$, i.e. the form which is the canonical symmetric bilinear form on the even elements $\mathbb{R}^{N}$, is the canonical [[symplectic form]] on the odd elements in $\mathbb{R}^{0|2p}$ and is zero on mixed pairs of elements.

The corresponding [[super Lie algebras]] are called the \emph{orthosymplectic Lie algebras} $\mathfrak{osp}(N|2p)$. Over a [[field]] of [[characteristic zero]] these constitute the infinite $B$- and $D$-series in the \href{super%20Lie%20algebra#Classification}{classification of simple super Lie algebras}. They are closely related to [[superconformal]] symmetry (e.g. \hyperlink{DAuriaFerrareLledoVaradarajan00}{D'Auria-Ferrara-Lledo-Varadarajan 00}), see at \emph{\href{supersymmetry#ClassificationSuperconformal}{supersymmetry -- Classification -- Superconformal symmetry}}.

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

\begin{itemize}%
\item $\mathfrak{osp}(1\vert 32)$ is a limiting case of the [[M-theory super Lie algebra]] (\hyperlink{FernandezIzquierdoOlmo15}{Fernandez-Izquierdo-Olmo 15}).

Further relation also to the [[type II supersymmetry algebra]] had been discussed in (\hyperlink{BergshoessProeyen00}{Bergshoess-Proeyen 00}).


\item The [[M-theory super Lie algebra]] is actually a sub-super Lie algebra of $\mathfrak{osp}(1\vert 64)$ (\hyperlink{vanHoltenVanProeyen82}{vanHolten-VanProeyen 82}, \hyperlink{BarsDelidumanMinic99}{Bars-Deliduma-nMinic 99, (11)-(15)}, \hyperlink{West00}{West 00, section 7})

In fact $OSp(1\vert 64)$ is the smallest simple supergroup that contains the [[M-theory super Lie algebra]]. Moreover, it is generated from its M-theory sub-algebra and the [[special conformal transformations]] (\hyperlink{BarsDelidumanMinic99}{Bars-Deliduman-Minic 99, p. 5})


\item The [[super anti de Sitter spacetimes]] are quotients of orthosymplectic super Lie groups:



\end{itemize}
\begin{tabular}{l|l}
$\phantom{A}$$d$$\phantom{A}$&$\phantom{A}$[[super anti de Sitter spacetime]]$\phantom{A}$\\
\hline 
$\phantom{A}$4$\phantom{A}$&$\;\;\;\;\frac{OSp(8\vert4)}{Spin(3,1) \times SO(7)}\;\;\;\;$\\
$\phantom{A}$5$\phantom{A}$&$\;\;\;\;\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}\;\;\;\;$\\
$\phantom{A}$7$\phantom{A}$&$\;\;\;\;\frac{OSp(6,2 \vert 4)}{Spin(6,1) \times SO(4)}\;\;\;\;$\\
\end{tabular}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[orthogonal Lie algebra]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Victor Kac]], pages 9-10 of \emph{A sketch of Lie superalgebra theory}, Comm. Math. Phys. Volume 53, Number 1 (1977), 31-64. (\href{https://projecteuclid.org/euclid.cmp/1103900590}{EUCLID})


\item [[Manfred Scheunert]], chapter II, 4.3.A of \emph{The theory of Lie superalgebras. An introduction}, Lect. Notes Math. 716 (1979)


\item Richard Joseph Farmer, \emph{Orthosymplectic superalgebras in mathematics and science}, PhD Thesis (1984) (\href{http://eprints.utas.edu.au/19542/}{web}, \href{http://eprints.utas.edu.au/19542/1/whole_FarmerRichardJoseph1985_thesis.pdf}{pdf})


\item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], section II.2.2 in \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991)


\item [[Riccardo D'Auria]], S. Ferrara, M. A. Lled\'o{}, [[Veeravalli Varadarajan]], \emph{Spinor Algebras}, J.Geom.Phys. 40 (2001) 101-128 (\href{http://arxiv.org/abs/hep-th/0010124}{arXiv:hep-th/0010124})


\item N. A. Gromov, I. V. Kostyakov, V. V. Kuratov, \emph{Cayley-Klein contractions of orthosymplectic superalgebras} (\href{http://arxiv.org/abs/hep-th/0110257}{arXiv:hep-th/0110257})


\item Steven V Sam, \emph{Orthosymplectic Lie superalgebras, Koszul duality, and a complete intersection analogue of the Eagon-Northcott complex} (\href{http://arxiv.org/abs/1312.2255}{arXiv:1312.2255})


\item G.I. Lehrer, R.B. Zhang, \emph{The second fundamental theorem of invariant theory for the orthosymplectic supergroup} (\href{http://arxiv.org/abs/1407.1058}{arXiv:1407.1058})



\end{itemize}
\hypertarget{in_superstring_theory}{}\subsubsection*{{In superstring theory}}\label{in_superstring_theory}

\begin{itemize}%
\item [[Igor Bandos]], Jerzy Lukierski, Christian Preitschopf, [[Dmitri Sorokin]], \emph{OSp supergroup manifolds, superparticles and supertwistors}, Phys.Rev.D61:065009, 2000 (\href{https://arxiv.org/abs/hep-th/9907113}{arXiv:hep-th/9907113})


\item [[Eric Bergshoeff]], [[Antoine Van Proeyen]], \emph{The many faces of $OSp(1|32)$}, Class. Quantum Grav. 17 (2000) 3277--3303 (\href{http://arxiv.org/abs/hep-th/0003261}{arXiv:hep-th/0003261})


\item [[Horatiu Nastase]], \emph{Towards a Chern-Simons M theory of $OSp(1\vert 32) \times OSp(1\vert 32)$} (\href{https://arxiv.org/abs/hep-th/0306269}{arXiv:hep-th/0306269})


\item J.J. Fernandez, J.M. Izquierdo, M.A. del Olmo, \emph{Contractions from $osp(1|32) \oplus osp(1|32)$ to the M-theory superalgebra extended by additional fermionic generators}, Nuclear Physics B Volume 897, August 2015, Pages 87--97 (\href{http://arxiv.org/abs/1504.05946}{arXiv:1504.05946})


\item [[Jan-Willem van Holten]], [[Antoine Van Proeyen]], \emph{$N=1$ Supersymmetry Algebras in $D=2$, $D=3$, $D=4$ $MOD-8$}, J.Phys. A15 (1982) 3763 (\href{http://inspirehep.net/record/177060}{spire:177060})


\item [[Itzhak Bars]], C. Deliduman, D. Minic, \emph{Lifting M-theory to Two-Time Physics}, Phys.Lett. B457 (1999) 275-284 (\href{https://arxiv.org/abs/hep-th/9904063}{arXiv:hep-th/9904063})


\item [[Itzhak Bars]], \emph{2T Physics Formulation of Superconformal Dynamics Relating to Twistors and Supertwistors}, Phys.Lett. B483 (2000) 248-256 (\href{https://arxiv.org/abs/hep-th/0004090}{arXiv:hep-th/0004090})


\item [[Peter West]], \emph{Hidden Superconformal Symmetry in M Theory}, JHEP 0008:007, 2000 (\href{https://arxiv.org/abs/hep-th/0005270}{arXiv:hep-th/0005270})



\end{itemize}
\hypertarget{representation_theory}{}\subsubsection*{{Representation theory}}\label{representation_theory}

The [[representation theory]] ([[singleton representation|singleton representations]]) of the orthosymplectic group is discussed in

\begin{itemize}%
\item [[Hermann Nicolai]], [[Ergin Sezgin]], \emph{Singleton representations of $Osp(N,4)$}, Physics Letters B, Volume 143, Issues 4--6, 16 August 1984, Pages 389-395

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

[[!redirects orthosymplectic super Lie algebras]]

[[!redirects orthosymplectic supergroup]] [[!redirects orthosymplectic supergroups]]

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

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



\end{document}