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\begin{document}

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\section*{super Lie algebra}

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

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

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

\hypertarget{superalgebra_and_supergeometry}{}\paragraph*{{Super-Algebra and Super-Geometry}}\label{superalgebra_and_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{as_lie_algebras_internal_to_super_vector_spaces}{As Lie algebras internal to super vector spaces}\dotfill \pageref*{as_lie_algebras_internal_to_super_vector_spaces} \linebreak
\noindent\hyperlink{as_supergraded_lie_algebras}{As super-graded Lie algebras}\dotfill \pageref*{as_supergraded_lie_algebras} \linebreak
\noindent\hyperlink{as_formal_duals_of_a_chevalleyeilenberg_superalgebras}{As formal duals of a Chevalley-Eilenberg super-algebras}\dotfill \pageref*{as_formal_duals_of_a_chevalleyeilenberg_superalgebras} \linebreak
\noindent\hyperlink{AsSuperRepresentableLieAlgebrasOverSuperpoints}{As super-representable Lie algebras in the topos over superpoints}\dotfill \pageref*{AsSuperRepresentableLieAlgebrasOverSuperpoints} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Classification}{Classification}\dotfill \pageref*{Classification} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{super Lie algebra} is the analog of a [[Lie algebra]] in [[superalgebra]]/[[supergeometry]].

See also [[supersymmetry]].

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

There are various equivalent ways to state the definition of super Lie algebras. Here are a few (for more discussion see at \emph{[[geometry of physics -- superalgebra]]}):

\hypertarget{as_lie_algebras_internal_to_super_vector_spaces}{}\subsubsection*{{As Lie algebras internal to super vector spaces}}\label{as_lie_algebras_internal_to_super_vector_spaces}

\begin{defn}
\label{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces}\hypertarget{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces}{}
A \emph{[[super Lie algebra]]} is a [[Lie algebra]] [[internalization|internal]] to the [[symmetric monoidal category]] $sVect = (Vect^{\mathbb{Z}/2}, \otimes_k, \tau^{super} )$ of [[super vector spaces]]. Hence this is

\begin{enumerate}%
\item a [[super vector space]] $\mathfrak{g}$;


\item a homomorphism

\begin{displaymath}
[-,-] \;\colon\; \mathfrak{g} \otimes_k \mathfrak{g} \longrightarrow \mathfrak{g}
\end{displaymath}
of super vector spaces (the \emph{super Lie bracket})



\end{enumerate}
such that

\begin{enumerate}%
\item the bracket is skew-symmetric in that the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    \mathfrak{g} \otimes_k \mathfrak{g}
      &
    \overset{\tau^{super}_{\mathfrak{g},\mathfrak{g}}}{\longrightarrow}
      &
    \mathfrak{g} \otimes_k \mathfrak{g}
    \\
    {}^{\mathllap{[-,-]}}\downarrow && \downarrow^{\mathrlap{[-,-]}}
    \\
    \mathfrak{g} &\underset{-1}{\longrightarrow}& \mathfrak{g}
  }
\end{displaymath}
(here $\tau^{super}$ is the [[braiding]] [[natural isomorphism]] in the category of [[super vector spaces]])


\item the [[Jacobi identity]] holds in that the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g}
    &&
      \overset{\tau^{super}_{\mathfrak{g}, \mathfrak{g}} \otimes_k id }{\longrightarrow}
    &&
    \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g}
    \\
    & {}_{\mathllap{\left[-,\left[-,-\right]\right]} - \left[\left[-,-\right],-\right] }\searrow && \swarrow_{\mathrlap{\left[-,\left[-,-\right]\right]}}
    \\
    && \mathfrak{g}
  }
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\hypertarget{as_supergraded_lie_algebras}{}\subsubsection*{{As super-graded Lie algebras}}\label{as_supergraded_lie_algebras}

Externally this means the following:

\begin{prop}
\label{SuperLieAlgebraTraditional}\hypertarget{SuperLieAlgebraTraditional}{}
A [[super Lie algebra]] according to def. \ref{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces} is equivalently

\begin{enumerate}%
\item a $\mathbb{Z}/2$-[[graded vector space]] $\mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}$;


\item equipped with a [[bilinear map]] (the \emph{super Lie bracket})

\begin{displaymath}
[-,-] : \mathfrak{g}\otimes_k \mathfrak{g} \to \mathfrak{g}
\end{displaymath}
which is \emph{graded} skew-symmetric: for $x,y \in \mathfrak{g}$ two elements of homogeneous degree $\sigma_x$, $\sigma_y$, respectively, then

\begin{displaymath}
[x,y] = -(-1)^{\sigma_x \sigma_y} [y,x]
  \,,
\end{displaymath}

\item that satisfies the $\mathbb{Z}/2$-graded [[Jacobi identity]] in that for any three elements $x,y,z \in \mathfrak{g}$ of homogeneous super-degree $\sigma_x,\sigma_y,\sigma_z\in \mathbb{Z}_2$ then

\begin{displaymath}
[x, [y, z]] = [[x,y],z] + (-1)^{\sigma_x \cdot \sigma_y} [y, [x,z]]
  \,.
\end{displaymath}


\end{enumerate}
A [[homomorphism]] of super Lie algebras is a homomorphisms of the underlying [[super vector spaces]] which preserves the Lie bracket. We write

\begin{displaymath}
sLieAlg
\end{displaymath}
for the resulting [[category]] of super Lie algebras.

\end{prop}
\hypertarget{as_formal_duals_of_a_chevalleyeilenberg_superalgebras}{}\subsubsection*{{As formal duals of a Chevalley-Eilenberg super-algebras}}\label{as_formal_duals_of_a_chevalleyeilenberg_superalgebras}

\begin{defn}
\label{CEAlgebraofSuperLieAlgebra}\hypertarget{CEAlgebraofSuperLieAlgebra}{}
For $\mathfrak{g}$ a [[super Lie algebra]] of [[finite number|finite]] [[dimension]], then its \emph{[[Chevalley-Eilenberg algebra]]} $CE(\mathfrak{g})$ is the super-[[Grassmann algebra]] on the [[dual vector space|dual]] super vector space

\begin{displaymath}
\wedge^\bullet  \mathfrak{g}^\ast
\end{displaymath}
equipped with a [[differential]] $d_{\mathfrak{g}}$ that on generators is the linear dual of the super Lie bracket

\begin{displaymath}
d_{\mathfrak{g}} \;\coloneqq\; [-,-]^\ast \;\colon\; \mathfrak{g}^\ast \to \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast
\end{displaymath}
and which is extended to $\wedge^\bullet \mathfrak{g}^\ast$ by the graded Leibniz rule (i.e. as a graded [[derivation]]).

$\,$

Here all elements are $(\mathbb{Z} \times \mathbb{Z}/2)$-bigraded, the first being the \emph{cohomological grading} $n$ in $\wedge^\n \mathfrak{g}^\ast$, the second being the \emph{super-grading} $\sigma$ (even/odd).

For $\alpha_i \in CE(\mathfrak{g})$ two elements of homogeneous bi-degree $(n_i, \sigma_i)$, respectively, the [[signs in supergeometry|sign rule]] is

\begin{displaymath}
\alpha_1 \wedge \alpha_2 = (-1)^{n_1 n_2} (-1)^{\sigma_1 \sigma_2}\; \alpha_2 \wedge \alpha_1
  \,.
\end{displaymath}
(See at \emph{[[signs in supergeometry]]} for discussion of this sign rule and of an alternative sign rule that is also in use. )

\end{defn}
We may think of $CE(\mathfrak{g})$ equivalently as the [[dg-algebra]] of [[invariant differential form|left-invariant]] [[super differential forms]] on the [[Lie theory|corresponding]] simply connected [[super Lie group]] .

The concept of [[Chevalley-Eilenberg algebras]] is traditionally introduced as a means to define [[Lie algebra cohomology]]:

\begin{defn}
\label{SuperLieAlgebraCohomology}\hypertarget{SuperLieAlgebraCohomology}{}
Given a [[super Lie algebra]] $\mathfrak{g}$, then

\begin{enumerate}%
\item an \emph{$n$-cocycle} on $\mathfrak{g}$ (with [[coefficients]] in $\mathbb{R}$) is an element of degree $(n,even)$ in its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ (def. \ref{CEAlgebraofSuperLieAlgebra}) which is $d_{\mathbb{g}}$ closed.


\item the cocycle is non-trivial if it is not $d_{\mathfrak{g}}$-exact


\item hene the \emph{super-[[Lie algebra cohomology]]} of $\mathfrak{g}$ (with [[coefficients]] in $\mathbb{R}$) is the [[cochain cohomology]] of its [[Chevalley-Eilenberg algebra]]

\begin{displaymath}
H^\bullet(\mathfrak{g}, \mathbb{R}) = H^\bullet(CE(\mathfrak{g}))
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
The following says that the [[Chevalley-Eilenberg algebra]] is an equivalent incarnation of the [[super Lie algebra]]:

\begin{prop}
\label{}\hypertarget{}{}
The functor

\begin{displaymath}
CE \;\colon\; sLieAlg^{fin}  \hookrightarrow dgAlg^{op}
\end{displaymath}
that sends a finite dimensional [[super Lie algebra]] $\mathfrak{g}$ to its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ (def. \ref{CEAlgebraofSuperLieAlgebra}) is a [[fully faithful functor]] which hence exibits [[super Lie algebras]] as a [[full subcategory]] of the [[opposite category]] of [[differential-graded algebras]].

\end{prop}
\hypertarget{AsSuperRepresentableLieAlgebrasOverSuperpoints}{}\subsubsection*{{As super-representable Lie algebras in the topos over superpoints}}\label{AsSuperRepresentableLieAlgebrasOverSuperpoints}

Equivalently, a super Lie algebra is a ``super-representable'' Lie algebra [[internalization|internal]] to the [[cohesive (∞,1)-topos]] [[Super∞Grpd]] over the site of [[super points]] (\hyperlink{Sachse08}{Sachse 08, Section 3.2, towards cor. 3.3}).

See the discussion at [[superalgebra]] for details on this.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Classification}{}\subsubsection*{{Classification}}\label{Classification}

(\href{Kac77a}{Kac 77a}, \hyperlink{Kac77b}{Kac 77b}) states a classification of super Lie algebras which are

\begin{enumerate}%
\item finite dimensional


\item simple


\item over a [[field]] of [[characteristic zero]].



\end{enumerate}
Such an algebra is called of \emph{classical type} if the [[action]] of its even-degree part on the odd-degree part is completely reducible. Those simple finite dimensional algebras not of classical type are of \emph{Cartan type}.

\begin{enumerate}%
\item classical type

\begin{enumerate}%
\item four infinite series

\begin{enumerate}%
\item $A(m,n)$


\item $B(m,n) =$ [[osp]]$(2m+1,2n)$ $m\geq 0$, $n \gt 0$


\item $C(n)$


\item $D(m,n) =$[[osp]]$(2m,2n)$ $m \geq 2$, $n \gt 0$



\end{enumerate}

\item two exceptional ones

\begin{enumerate}%
\item $F(4)$


\item $G(3)$



\end{enumerate}

\item a family $D(2,1;\alpha)$ of deformations of $D(2,1)$


\item two ``strange'' series

\begin{enumerate}%
\item $P(n)$


\item $Q(n)$



\end{enumerate}


\end{enumerate}

\item Cartan type

(\ldots{})



\end{enumerate}
The underlying even-graded [[Lie algebra]] for type 2 is as follows

\begin{tabular}{l|l|l}
$\mathfrak{g}$&$\mathfrak{g}_{even}$&$\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{odd}$\\
\hline 
$B(m,n)$&$B_m \oplus C_n$&vector $\otimes$ vector\\
$D(m,n)$&$D_m \oplus C_n$&vector $\otimes$ vector\\
$D(2,1,\alpha)$&$A_1 \oplus A_1 \oplus A_1$&vector $\otimes$ vector $\otimes$ vector\\
$F(4)$&$B_3\otimes A_1$&spinor $\otimes$ vector\\
$G(3)$&$G_2\oplus A_1$&spinor $\otimes$ vector\\
$Q(n)$&$A_n$&adjoint\\
\end{tabular}

For type 1 the $\mathbb{Z}/2\mathbb{Z}$-grading lifts to an $\mathbb{Z}$-grading with $\mathfrak{g} = \mathfrak{g}_{-1}\oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$.

\begin{tabular}{l|l|l}
$\mathfrak{g}$&$\mathfrak{g}_{even}$&$\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{{-1}}$\\
\hline 
$A(m,n)$&$A_m \oplus A_n \oplus C$&vector $\otimes$ vector $\otimes$ $\mathbb{C}$\\
$A(m,m)$&$A_m \oplus A_n$&vector $\otimes$ vector\\
$C(n)$&$\mathbb{C}_{-1} \oplus \mathbb{C}$&vector $\otimes$ $\mathbb{C}$\\
\end{tabular}

reviewed e.g. in (\hyperlink{Farmer84}{Farmer 84, p. 25,26}, \hyperlink{Minwalla98}{Minwalla 98, section 4.1}).

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

Some obvious but important classes of examples are the following:

\begin{example}
\label{SuperVectorSpaceAsAbelianSuperLieAlgebra}\hypertarget{SuperVectorSpaceAsAbelianSuperLieAlgebra}{}
every $\mathbb{Z}/2$-[[graded vector space]] $V$ becomes a [[super Lie algebra]] (def. \ref{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces}, prop. \ref{SuperLieAlgebraTraditional}) by taking the super Lie bracket to be the [[zero morphism|zero map]]

\begin{displaymath}
[-,-] = 0
  \,.
\end{displaymath}
These may be called the ``abelian'' super Lie algebras.

\end{example}
\begin{example}
\label{OrdinaryLieAlgebraAsSuperLieAlgebra}\hypertarget{OrdinaryLieAlgebraAsSuperLieAlgebra}{}
Every ordinary [[Lie algebras]] becomes a [[super Lie algebra]] (def. \ref{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces}, prop. \ref{SuperLieAlgebraTraditional}) concentrated in even degrees. This constitutes a [[fully faithful functor]]

\begin{displaymath}
LieAlg \hookrightarrow sLieAlg
  \,.
\end{displaymath}
which is a [[coreflective subcategory]] inclusion in that it has a [[left adjoint]]

\begin{displaymath}
LieAlg
    \underoverset
     {\underset{ \overset{ \rightsquigarrow}{(-)} }{\longleftarrow}}
     {\hookrightarrow}
     {\bot}
  sLieAlg
\end{displaymath}
given on the underlying super vector spaces by restriction to the even graded part

\begin{displaymath}
\overset{\rightsquigarrow}{\mathfrak{s}} = \mathfrak{s}_{even}
  \,.
\end{displaymath}
\end{example}
\begin{itemize}%
\item The [[super Poincare Lie algebra]] and various of its polyvector extension are super-extension of the ordinary [[Poincare Lie algebra]]. These are the [[supersymmetry algebras]] in the strict original sense of the word. For more on this see at \emph{[[geometry of physics -- supersymmetry]]}.


\item [[super q-Schur algebra]]


\item For every [[pointer topological space]], the [[Whitehead product]] makes its [[homotopy groups]] into a super Lie algebra over the [[ring]] of [[integers]].


\item higher super Lie algebras

Just as [[Lie algebras]] are [[vertical categorification|categorified]] to [[L-infinity algebra]]s and [[L-infinity algebroid]]s, so super Lie algebras categorifie to [[super L-infinity algebra]]s. A secretly famous example is the

\begin{itemize}%
\item [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]]

\end{itemize}


\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Haag–Łopuszański–Sohnius theorem]]


\item [[geometry of physics -- supersymmetry]]



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

According to \hyperlink{Kac77b}{Kac77b} the definition of super Lie algebra is originally due to

\begin{itemize}%
\item [[Felix Berezin]], G. I. Kac, Math. Sbornik 82, 343---351 (1970) (Russian)

\end{itemize}
The original references on the classification of super Lie algebras are

\begin{itemize}%
\item [[Victor Kac]], \emph{Lie superalgebras}, Advances in Math. 26 (1977), no. 1, 8--96.


\item [[Victor Kac]], \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})



\end{itemize}
See also

\begin{itemize}%
\item [[Werner Nahm]], V. Rittenberg, [[Manfred Scheunert]], \emph{The classification of graded Lie algebras} , Physics Letters B Volume 61, Issue 4, 12 April 1976, Pages 383--384 (\href{http://www.sciencedirect.com/science/article/pii/0370269376905943}{publisher})


\item M. Parker, \emph{Classification Of Real Simple Lie Superalgebras Of Classical Type}, J.Math.Phys. 21 (1980) 689-697 (\href{http://inspirehep.net/record/157627?ln=en}{spire})



\end{itemize}
Further discussion of classification related specifically to \hyperlink{supersymmetry#Classification}{classification of supersymmetry} is due to

\begin{itemize}%
\item [[Werner Nahm]], \emph{[[Supersymmetries and their Representations]]}, Nucl.Phys. B135 (1978) 149 (\href{https://inspirehep.net/record/120988/}{spire}, \href{http://cds.cern.ch/record/132743/files/197709213.pdf}{pdf})


\item [[Steven Shnider]], \emph{The superconformal algebra in higher dimensions}, Letters in Mathematical Physics November 1988, Volume 16, Issue 4, pp 377-383


\item [[Victor Kac]], \emph{Classification of supersymmetries}, Proceedings of the ICM, Beijing 2002, vol. 1, 319--344 (\href{http://arxiv.org/abs/math-ph/0302016}{arXiv:math-ph/0302016})



\end{itemize}
Introductions and surveys include

\begin{itemize}%
\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 L. Frappat, A. Sciarrino, P. Sorba, \emph{Dictionary on Lie Superalgebras} (\href{http://arxiv.org/abs/hep-th/9607161}{arXiv:hep-th/9607161})


\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 L. Frappat, A. Sciarrino, P. Sorba, \emph{Dictionary on Lie Superalgebras} (\href{http://arxiv.org/abs/hep-th/9607161}{arXiv:hep-th/9607161})


\item D. Leites, \emph{Lie superalgebras}, J. Soviet Math. 30 (1985), 2481--2512 (\href{http://dx.doi.org/10.1007/BF02249121}{web})


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


\item D. Westra, \emph{Superrings and supergroups} (\href{http://www.mat.univie.ac.at/~michor/westra_diss.pdf}{pdf})


\item [[Shiraz Minwalla]], \emph{Restrictions imposed by superconformal invariance on quan tum field theories} Adv. Theor. Math. Phys. 2, 781 (1998) (\href{http://arxiv.org/abs/hep-th/9712074}{arXiv:hep-th/9712074}).



\end{itemize}
Discussion in the topos over superpoints is in

\begin{itemize}%
\item [[Christoph Sachse]], \emph{A Categorical Formulation of Superalgebra and Supergeometry} (\href{http://arxiv.org/abs/0802.4067}{arXiv:0802.4067})

\end{itemize}
Discussion of [[Lie algebra extensions]] for super Lie algebras includes

\begin{itemize}%
\item [[Dmitri Alekseevsky]], [[Peter Michor]], Wolfgang Ruppert, \emph{Extensions of super Lie algebras}, J. Lie Theory 15 (2005) No. 1, 125--134 (\href{http://arxiv.org/abs/math/0101190}{arXiv:math/0101190})

\end{itemize}
[[!redirects Lie superalgebra]] [[!redirects super Lie algebras]]

[[!redirects super Lie bracket]] [[!redirects super Lie brackets]]



\end{document}