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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{super Poincaré Lie algebra} [[!redirects super Poincare Lie algebra]] \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory} [[!include infinity-Lie theory - contents]] \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{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{LieAlgebraCohomology}{Lie algebra cohomology}\dotfill \pageref*{LieAlgebraCohomology} \linebreak \noindent\hyperlink{extensions}{Extensions}\dotfill \pageref*{extensions} \linebreak \noindent\hyperlink{super_algebra_extensions}{Super $L_\infty$-algebra extensions}\dotfill \pageref*{super_algebra_extensions} \linebreak \noindent\hyperlink{PolyvectorExtensions}{Extended super Poincar\'e{} Lie algebra -- Polyvector extensions}\dotfill \pageref*{PolyvectorExtensions} \linebreak \noindent\hyperlink{as_current_algebras_of_the_gs_super_branes}{As current algebras of the GS super $p$-branes}\dotfill \pageref*{as_current_algebras_of_the_gs_super_branes} \linebreak \noindent\hyperlink{PolyvectorExtensionsAsAutomorphismLieAlgebras}{As automorphism Lie algebras of Lie $n$-superalgebras}\dotfill \pageref*{PolyvectorExtensionsAsAutomorphismLieAlgebras} \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{PolyvectorExtensionRefs}{Polyvector extensions}\dotfill \pageref*{PolyvectorExtensionRefs} \linebreak \noindent\hyperlink{ReferencesLieAlgebraCohomology}{Super Lie algebra cohomology}\dotfill \pageref*{ReferencesLieAlgebraCohomology} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{super Poincar\'e{} Lie algebra} is a [[super Lie algebra]] extension of a [[Poincaré Lie algebra]]. The corresponding [[super Lie group]] is the [[super Euclidean group]] (except for the signature of the metric). \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} Let $d \in \mathbb{N}$ and consider [[Minkowski spacetime]] $\mathbb{R} ^{d-1,1}$ of [[dimension]] $d$. For $Spin(d-1,1)$ the corresponding [[spin group]], let \begin{displaymath} S \in Rep(Spin(d-1,1)) \end{displaymath} be a real [[spin representation]] ([[Majorana spinor]]), which has the property (\href{Majorana+spinor#SpinorToVectorBilinearPairing}{def.}, \href{Majorana+spinor#SpinorToVectorPairing}{prop.}) that there exists a [[linear map]] \begin{displaymath} \Gamma \;\colon\; S \otimes S \longrightarrow \mathbb{R}^{d-1,1} \end{displaymath} which is \begin{enumerate}% \item symmetric: \item $Spin(V)$ [[equivariance|equivariant]] (a [[homomorphism]] of [[spin representations]]). \end{enumerate} For a classification of spin representations with this property see at \emph{[[spin representations]]} the sections \emph{\href{spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature}{real irreducible spin representations in Lorentz signature}} and \emph{\href{spin+representation#SuperPoincareBrackets}{super Poincar\'e{} brackets}}. For explicit construction in components see at \emph{[[Majorana spinor]]} the section \emph{\href{Majorana+spinor#TheSpinorPairingToVectors}{The spinor pairing to vectors}}. \begin{defn} \label{SuperPoincare}\hypertarget{SuperPoincare}{} The \textbf{super Poincar\'e{} Lie algebra} $\mathfrak{siso}_S(d-1,1)$ of $d$-dimensional [[Minkowski spacetime]] with respect to the [[spin representation]] $S$ with symmetric and $Spin(V)$-equivariant pairing $\Gamma \colon S \otimes S \to \mathbb{R}^{d-1,1}$ is the [[super Lie algebra|super]] [[Lie algebra extension]] of the [[Poincaré Lie algebra]] by $\Pi S$ (the [[vector space]] underlying $S$ taken in odd degree) \begin{displaymath} \Pi S \longrightarrow \mathfrak{siso}_S(d-1,1) \longrightarrow \mathfrak{iso}(d-1,1) \simeq \mathbb{R}^{d-1,1} \ltimes \mathfrak{so}(d-1,1) \,, \end{displaymath} where the [[Lie bracket]] of elements in $\mathfrak{so}(d-1,1)$ with those in $S$ is the given [[action]], the Lie bracket of elements of $\mathbb{R}^{d-1,1}$ with those on $S$ is trivial, and the Lie bracket of two elements $s_1, s_2 \in S$ is given by $\Gamma$: \begin{displaymath} [s_1,s_2] \coloneqq \Gamma(s_1,s_2) \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} It is precisely the symmetry and $Spin(V)$-equivariant assumption on $\Gamma$ that makes this a well defined [[super Lie algebra]]: the symmetry corresponds to the graded skew-symmetry of the [[Lie bracket]] on elements in $S$, which are regarded as odd, and the $Spin(V)$-equivariance yields the nontrivial [[Jacobi identity]] for $o \in \mathfrak{so}(d-1,1)$ and $s_1, s_2 \in S$: \begin{displaymath} \Gamma([o,s_1], s_2) + \Gamma(s_1, [o,s_2]) = [o, \Gamma(s_1,s_2)] \,. \end{displaymath} \end{remark} \begin{remark} \label{CEAlgebraOfSuperPoincare}\hypertarget{CEAlgebraOfSuperPoincare}{} By the general discussion at [[Chevalley-Eilenberg algebra]], we may characterize the [[super Poincaré Lie algebra]] $\mathfrak{siso}_S(D-1,1)$ by its CE super-[[dg-algebra]] $CE(\mathfrak{siso}_S(D-1,1))$ ``of [[left-invariant 1-forms]]'' on its group manifold. Write $\{\omega_a{}^b\}_{a,b}$ for the canonical basis of the [[special orthogonal Lie algebra|special orthogonal]] [[matrix Lie algebra]] $\mathfrak{so}(D-1,1)$ and write $\{\psi_\alpha\}_\alpha$ for a corresponding [[basis]] of the [[spin representation]] $S$. The [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{siso}_N(d-1,1))$ is generated on \begin{itemize}% \item elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$ \item and elements $\{\psi^\alpha\}$ of degree $(1,odd)$ \end{itemize} with the differential defined by \begin{displaymath} d_{CE} \omega^{a b} = \omega^a{}_b \wedge \omega^{b c} \end{displaymath} \begin{displaymath} d_{CE} e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi \end{displaymath} \begin{displaymath} d_{CE} \psi = \frac{1}{4} \omega^{ a b} \wedge \Gamma_{a b} \psi \,. \end{displaymath} \end{remark} \begin{remark} \label{}\hypertarget{}{} Removing all terms involving $\omega$ here yields the [[Chevalley-Eilenberg algebra]] of the [[super translation algebra]] $\mathbb{R}^{D;N}$. \end{remark} \begin{remark} \label{}\hypertarget{}{} The abstract generators in def. \ref{CEAlgebraOfSuperPoincare} are identified with [[left invariant 1-forms]] on the [[super-translation group]] as follows. Let $(x^a, \theta^\alpha)$ be the canonical [[coordinates]] on the [[supermanifold]] $\mathbb{R}^{d|N}$ underlying the super translation group. Then the identification is \begin{itemize}% \item $\psi^\alpha = d \theta^\alpha$. \item $e^a = d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta$. \end{itemize} This then gives the formula for the differential of the super-[[vielbein]] in def. \ref{CEAlgebraOfSuperPoincare} as \begin{displaymath} \begin{aligned} d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta) \\ & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta \\ & = \frac{i}{2} \overline{\psi}\Gamma^a \psi \end{aligned} \,. \end{displaymath} \end{remark} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{LieAlgebraCohomology}{}\subsubsection*{{Lie algebra cohomology}}\label{LieAlgebraCohomology} The super Poincar\'e{} Lie algebra has, on top of the [[Lie algebra cohomology|Lie algebra cocycles]] that it inherits from $\mathfrak{so}(n)$, a discrete number of exceptiona cocycles bilinear in the spinors, on the [[super translation algebra]], that exist only in very special dimensions. The following theorem has been stated at various placed in the physics literature (known there as the \emph{[[brane scan]]} for $\kappa$-symmetry in \emph{[[Green-Schwarz action functionals]]} for super-$p$-[[branes]] on [[super-Minkowski spacetime]]). A full proof is in \hyperlink{Brandt12-13}{Brandt 12-13}. The following uses the notation in terms of [[division algebras]] (\hyperlink{BaezHuerta10}{Baez-Huerta 10}). \textbf{Theorem} \begin{itemize}% \item In dimensional $d = 3,4,6, 10$, $\mathfrak{siso}(d-1,1)$ has a nontrivial 3-cocycle given by \begin{displaymath} (\psi, \phi, A) \mapsto g(\psi \cdot \phi, A) \end{displaymath} for spinors $\psi, \phi \in \mathcal{S}$ and vectors $A \in \mathcal{T}$, and 0 otherwise. \item In dimensional $d = 4,5,7, 11$, $\mathfrak{siso}(d-1,1)$ has a nontrivial 4-cocycle given by \begin{displaymath} (\Psi, \Phi, \mathcal{A}, \mathcal{B}) \mapsto \langle \Psi , (\mathcal{A}\mathcal{B}- \mathcal{B} \mathcal{A})\Phi \rangle \end{displaymath} for spinors $\Psi, \Phi \in \mathcal{S}$ and vectors $\mathcal{A}, \mathcal{B} \in \mathcal{V}$, with the commutator taken in the Clifford algebra. \end{itemize} The 4-cocycle in $d = 11$ is the one that induces the [[supergravity Lie 3-algebra]]. All these cocycles are controled by the relevant [[Fierz identities]]. \hypertarget{extensions}{}\subsubsection*{{Extensions}}\label{extensions} \hypertarget{super_algebra_extensions}{}\paragraph*{{Super $L_\infty$-algebra extensions}}\label{super_algebra_extensions} The [[super L-infinity algebra]] [[infinity-Lie algebra cohomology]] of the super Poincar\'e{} Lie algebra corresponding to the \hyperlink{LieAlgebraCohomology}{above} cocycles involves [[supergravity Lie 6-algebra]] $\to$ [[supergravity Lie 3-algebra]] $\to$ \textbf{super-Poincar\'e{} Lie algebra} \hypertarget{PolyvectorExtensions}{}\paragraph*{{Extended super Poincar\'e{} Lie algebra -- Polyvector extensions}}\label{PolyvectorExtensions} The super-Poincar\'e{} Lie algebra has a class of super [[Lie algebra extensions]] called \emph{[[extended supersymmetry]]} algebras or \emph{polyvector extensions} , because they involve additional generators that transforn as skew-symmetric [[tensors]]. A complete classification is in (\hyperlink{ACDP}{ACDP}). For instance the ``[[M-theory Lie algebra]]'' is a polyvector extension of the super Poincar\'e{} Lie algebra $\mathfrak{siso}_{N=1}(10,1)$ by polyvectors of rank $p = 2$ and $p=5$ (the [[M2-brane]] and the [[M5-brane]] in the [[brane scan]]), see below \hyperlink{PolyvectorExtensionsAsAutomorphismLieAlgebras}{Polyvector extensions as automorphism Lie algebras}. \hypertarget{as_current_algebras_of_the_gs_super_branes}{}\paragraph*{{As current algebras of the GS super $p$-branes}}\label{as_current_algebras_of_the_gs_super_branes} The polyvector extensions arise as the super Lie algebras of [[conserved currents]] of the [[Green-Schwarz super p-brane sigma-models]] (\hyperlink{AGIT89}{AGIT 89}). \hypertarget{PolyvectorExtensionsAsAutomorphismLieAlgebras}{}\paragraph*{{As automorphism Lie algebras of Lie $n$-superalgebras}}\label{PolyvectorExtensionsAsAutomorphismLieAlgebras} At least some of the \hyperlink{PolyvectorExtensions}{polyvector extensions} of the super Poincar\'e{} Lie algebra arise as the [[automorphism]] super Lie algebras of the [[Lie n-algebra]] [[infinity-Lie algebra cohomology|extensions]] classified by the cocycles discussed above. For instance the automorphisms of the [[supergravity Lie 3-algebra]] gives the ``[[M-theory Lie algebra]]''-extension of super-Poincar\'e{} in 11-dimensions (\hyperlink{FSS13}{FSS 13}). This is also discussed at \emph{\href{supergravity+Lie+3-algebra#Polyvector}{supergravity Lie 3-algebra -- Polyvector extensions}}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[super Minkowski spacetime]], [[signs in supergeometry]] \item [[super Poincaré group]], [[supersymmetry]] \item [[supermultiplet]], [[BPS state]] \item [[division algebra and supersymmetry]] \item [[R-symmetry]] \item [[super translation algebra]] \item [[Green-Schwarz action functional]], [[brane scan]] \item [[4d supergravity Lie 2-algebra]], [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Introducing the [[super Poincaré Lie algebra]] (``[[supersymmetry]]''): \begin{itemize}% \item [[Yuri Golfand]], [[Evgeny Likhtman]],\_On the Extensions of the Algebra of the Generators of the Poincaré Group by the Bispinor Generators\_, in: [[Victor Ginzburg]] et al. (eds.) \emph{I. E. Tamm Memorial Volume Problems of Theoretical Physics}, (Nauka, Moscow 1972), page 37, translated and reprinted in: [[Mikhail Shifman]] (ed.) \emph{[[The Many Faces of the Superworld]]} pp. 44-53, World Scientific (2000) (\href{https://doi.org/10.1142/9789812793850_0006}{doi:10.1142/9789812793850\_0006}) \end{itemize} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], section II.2.1 of \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991) \end{itemize} The seminal classification result of simple supersymmetry algebras is due to \begin{itemize}% \item [[Werner Nahm]], \emph{Supersymmetries and their Representations}, Nucl.Phys. B135 (1978) 149 (\href{http://inspirehep.net/record/120988?ln=en}{spire}) \end{itemize} Lecture notes include \begin{itemize}% \item \emph{Super spacetimes and super Poincar\'e{}-group} (\href{http://www.math.ucla.edu/~vsv/papers/ch6.pdf}{pdf}) \item [[Daniel Freed]], lecture 6 of \emph{Classical field theory and Supersymmetry}, IAS/Park City Mathematics Series Volume 11 (2001) (\href{https://www.ma.utexas.edu/users/dafr/pcmi.pdf}{pdf}) \item [[Daniel Freed]], \emph{Lecture 4 of [[Five lectures on supersymmetry]]} \item [[Veeravalli Varadarajan]], section 7 of \emph{[[Supersymmetry for mathematicians]]: An introduction}, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004) \end{itemize} See also \begin{itemize}% \item C. Chryssomalakos, [[José de Azcárraga]], J. M. Izquierdo and C. P\'e{}rez Bueno, \emph{The geometry of branes and extended superspaces} (\href{http://arxiv.org/abs/hep-th/9904137}{arXiv:hep-th/9904137}) \end{itemize} for discussion in the view of the [[brane scan]] and [[schreiber:The brane bouquet]] of super-$p$-[[brane]] [[Green-Schwarz sigma-models]]. \hypertarget{PolyvectorExtensionRefs}{}\subsubsection*{{Polyvector extensions}}\label{PolyvectorExtensionRefs} The Polyvector extensions of $\mathfrak{Iso}(\mathbb{R}^{10,1|32})$ (the ``[[M-theory super Lie algebra]]'') were first considered in \begin{itemize}% \item [[Riccardo D'Auria]], [[Pietro Fré]] \emph{[[GeometricSupergravity.pdf:file]]}, Nuclear Physics B201 (1982) 101-140 \item [[Jan-Willem van Holten]], [[Antoine Van Proeyen]], \emph{$N=1$ supersymmetry algebras in $d=2,3,4 \,mod\, 8$} J.Phys. A15, 3763 (1982). \end{itemize} Polyvector extensions were found as the algebra of [[conserved currents]] of the [[Green-Schwarz super p-branes]] in \begin{itemize}% \item [[José de Azcárraga]], [[Jerome Gauntlett]], J.M. Izquierdo, [[Paul Townsend]], \emph{Topological Extensions of the Supersymmetry Algebra for Extended Objects}, Phys.Rev.Lett. 63 (1989) 2443 (\href{https://inspirehep.net/record/26393?ln=en}{spire}) \end{itemize} reviewed in section 8.8. of \begin{itemize}% \item [[José de Azcárraga]], Jos\'e{} M. Izquierdo, \emph{Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics} , Cambridge monographs of mathematical physics, (1995) \end{itemize} and specifically for super-[[D-branes]] this discussion is in \begin{itemize}% \item Hanno Hammer, \emph{Topological Extensions of Noether Charge Algebras carried by D-p-branes}, Nucl.Phys. B521 (1998) 503-546 (\href{http://arxiv.org/abs/hep-th/9711009}{arXiv:hep-th/9711009}) \end{itemize} The role of polyvector extended supersymmetry algebras in [[supergravity]] and [[string theory]] is further highlighted in \begin{itemize}% \item [[Paul Townsend]], \emph{$p$-Brane Democracy} (\href{http://arxiv.org/abs/hep-th/9507048}{arXiv:hep-th/9507048}) \end{itemize} A comprehensive account and classification of the polyvector extensions of the super Poincar\'e{} Lie algebras is in \begin{itemize}% \item [[Dmiti Alekseevsky]], [[Vicente Cortés]], C. Devchand, [[Antoine Van Proeyen]], \emph{Polyvector Super-Poincar\'e{} Algebras} Commun.Math.Phys. 253 (2004) 385-422 (\href{http://arxiv.org/abs/hep-th/0311107}{arXiv:hep-th/0311107}) \end{itemize} \hypertarget{ReferencesLieAlgebraCohomology}{}\subsubsection*{{Super Lie algebra cohomology}}\label{ReferencesLieAlgebraCohomology} Discussion of the super-[[Lie algebra cohomology]] of the [[super Poincare Lie algebra]] goes back to work on [[Green-Schwarz sigma models]] in \begin{itemize}% \item [[José de Azcárraga]], [[Paul Townsend]], \emph{Superspace geometry and classification of supersymmetric extended objects}, Phys. Rev. Lett. 62, 2579--2582 (1989) \end{itemize} A rigorous classification of these cocycles was later given in \begin{itemize}% \item [[Friedemann Brandt]], \emph{Supersymmetry algebra cohomology} \emph{I: Definition and general structure} J. Math. Phys.51:122302, 2010, (\href{http://arxiv.org/abs/0911.2118}{arXiv:0911.2118}) \emph{II: Primitive elements in 2 and 3 dimensions}, J. Math. Phys. 51 (2010) 112303 (\href{http://arxiv.org/abs/1004.2978}{arXiv:1004.2978}) \emph{III: Primitive elements in four and five dimensions}, J. Math. Phys. 52:052301, 2011 (\href{http://arxiv.org/abs/1005.2102}{arXiv:1005.2102}) \emph{IV: Primitive elements in all dimensions from $D=4$ to $D=11$}, J. Math. Phys. 54, 052302 (2013) (\href{http://arxiv.org/abs/1303.6211}{arXiv:1303.6211}) \end{itemize} A classification of some special cases of signature/supersymmetry of this is also in the following (using a computer algebra system): \begin{itemize}% \item Michael Movshev, [[Albert Schwarz]], Renjun Xu, \emph{Homology of Lie algebra of supersymmetries} (\href{http://arxiv.org/abs/1011.4731}{arXiv:1011.4731}) \item Michael Movshev, [[Albert Schwarz]], Renjun Xu, \emph{Homology of Lie algebra of supersymmetries and of super Poincar\'e{} Lie algebra}, Nuclear Physics B Volume 854, Issue 2, 11 January 2012, Pages 483--503 (\href{http://arxiv.org/abs/1106.0335}{arXiv:1106.0335}) \end{itemize} For applications of this classification see also at \emph{[[Green-Schwarz action functional]]} and at \emph{[[brane scan]]}. An introduction to the exceptional fermionic cocycles on the super Poincar\'e{} Lie algebra, and their description using [[normed division algebras]], are discussed here: \begin{itemize}% \item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry I} (\href{http://arxiv.org/abs/0909.0551}{arXiv:0909.0551}) \item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry II} (\href{http://arxiv.org/abs/1003.3436}{arXiv:1003.34360}) \end{itemize} \begin{itemize}% \item [[John Huerta]], \emph{Division Algebras, Supersymmetry and Higher Gauge Theory}, (\href{http://arxiv.org/abs/1106.3385}{arXiv:1106.3385}) \end{itemize} This subsumes some of the results in (\hyperlink{AzcarragaTownsend89}{Azc\'a{}rraga-Townend}) Discussion of the corresponding [[super L-∞ algebra]] [[L-∞ extensions]] in the context of [[Green-Schwarz action functionals]] and [[schreiber:∞-Wess-Zumino-Witten theory]] is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]} \end{itemize} A direct constructions of ordinary (Lie algebraic) extensions of the super Poincar\'e{} Lie algebra by means of [[division algebras]] is in \begin{itemize}% \item Jerzy Lukierski, Francesco Toppan, \emph{Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory} (\href{http://cbpfindex.cbpf.br/publication_pdfs/NF00102.2010_08_03_10_47_48.pdf}{pdf}) \end{itemize} For more on this see at \emph{[[division algebra and supersymmetry]]}. [[!redirects super Poincare Lie algebras]] [[!redirects super Poincaré Lie algebra]] [[!redirects super Poincaré Lie algebras]] [[!redirects super Poincaré super Lie algebra]] [[!redirects super Poincaré super Lie algebras]] [[!redirects super-Poincaré Lie algebra]] \end{document}