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Tables-for-susy.tex
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\begin{document}
\raggedbottom
\noindent\textbf{\Large Tables for supersymmetry.}
\noindent{\scriptsize Bruno Le Floch, \'Ecole Normale Sup\'erieure, 2019.\\
Very sparse references, not always to original papers.\\
Help welcome at
\url{https://github.com/blefloch/tables-for-supersymmetry}
\par}
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\section[Lie algebras and groups]{Lie algebras and groups (dimension${}<\infty$)}
\subsection{Lie algebras}
\paragraph{Complex simple Lie algebras.}
Infinite series $\lie{A}_{n\geq 1}$, $\lie{B}_{n\geq 1}$, $\lie{C}_{n\geq 1}$, $\lie{D}_{n\geq 2}$ with
$\lie{A}_1=\lie{B}_1=\lie{C}_1$, $\lie{B}_2=\lie{C}_2$, $\lie{D}_2=\lie{A}_1\oplus \lie{A}_1$, $\lie{D}_3=\lie{A}_3$.
Five exceptions with dimensions
\begin{tabular}{|ccccc|}
$\lie{E}_6$ & $\lie{E}_7$ & $\lie{E}_8$ & $\lie{F}_4$ & $\lie{G}_2$ \\
$78$ & $133$ & $248$ & $52$ & $14$ \\[-1ex]
\end{tabular} .
% $\dim(\lie{E}_6)=78$, $\dim(\lie{E}_7)=133$, $\dim(\lie{E}_8)=248$, $\dim(\lie{F}_4)=52$, $\dim(\lie{G}_2)=14$.
\begin{tab}{*{3}{>{$}l<{$}}l}
\text{Type} & \text{Dimension} & \text{Lie algebra} \\\midrule
\lie{A}_n & n(n+2) & \lie{sl}(n+1,\CC) =\{\text{traceless}\} \\
\lie{B}_n & n(2n+1) & \lie{so}(2n+1,\CC) =\{\text{antisymmetric}\} \\
\lie{C}_n & n(2n+1) & \lie{sp}(2n,\CC)
=\left\{\left(\begin{smallmatrix}0&\one_n\\-\one_n&0\end{smallmatrix}\right)\times\text{symmetric}\right\}
\\
\lie{D}_n & n(2n-1) & \lie{so}(2n,\CC) =\{\text{antisymmetric}\} \\
\end{tab}
\paragraph{Roots and Weyl group.} The Weyl group has $\prod_i d_i$ elements where $d_i$~are degrees of fundamental invariants.
(Below, $\one_i$~denotes the $i$-th unit vector in $\ZZ^n$ and $1\leq i\neq j\leq n$.)
\begin{description}[topsep=0pt,parsep=0pt,partopsep=0pt,itemsep=0pt,leftmargin=1em]
\item[$\lie{A}_{n-1}$:] (note shifted rank) roots $\one_i-\one_j$, simple roots $\one_i-\one_{i+1}$.
The Weyl group~$S_n$ permutes the~$\one_i$.
Fundamental invariants: $x_1^k+\cdots+x_n^k$ for $2\leq k\leq n$.
\item[$\lie{B}_n$:] roots $\pm\one_i$ and $\pm\one_i\pm\one_j$, simple roots $\one_i-\one_{i+1}$ and $\one_n$.
The Weyl group $\{\pm 1\}^n\rtimes S_n$ permutes and changes signs of the~$\one_i$.
Fundamental invariants: $x_1^{2k}+\cdots+x_n^{2k}$ for $2\leq 2k\leq 2n$.
\item[$\lie{C}_n$:] roots $\pm2\one_i$ and $\pm\one_i\pm\one_j$, simple roots $\one_i-\one_{i+1}$ and $2\one_n$.
Same Weyl group and invariants as~$\lie{B}_n$.
\item[$\lie{D}_n$:] roots $\pm\one_i\pm\one_j$, simple roots $\one_i-\one_{i+1}$ and $\one_{n-1}+\one_n$.
The Weyl group $\{\pm 1\}^{n-1}\rtimes S_n$ permutes the~$\one_i$ and changes an even number of signs.
Fundamental invariants $x_1\cdots x_n$ and $x_1^{2k}+\cdots+x_n^{2k}$ for $2\leq 2k\leq 2n-2$.
\item[$\lie{E}_8$:] $\{\pm\one_i\pm\one_j\}\cup \{\frac{1}{2}\sum_{k=1}^8\epsilon_k\one_k\mid \epsilon_k=\pm 1, \prod_{k=1}^8\epsilon_k=-1\}$,
simple roots $\one_i-\one_{i+1}$ and $\frac{1}{2}(-\one_1-\cdots-\one_5+\one_6+\one_7+\one_8)$.
The $2^{14}\,3^5\,5^2\,7=696729600$-element Weyl group is $O_8^+(\FF_2)$.
Degrees of invariants are $\{d_i\}=\{2,8,12,14,18,20,24,30\}$, with mnemonic $1+(\text{primes from $7$ to $29$})$.
\item[$\lie{E}_7$:] roots $\sum_{i=1}^8 a_i \one_i$ of $\lie{E}_8$ with $a_1=\sum_{i=2}^8 a_i$,
simple roots are those of~$\lie{E}_8$ except $\one_1-\one_2$.
The $2^{10}\times 3^4\times 5\times 7=2903040$-element Weyl group is $\ZZ_2\times \Lie{PSp}_6(\FF_2)$. Degrees of invariants are $\{d_i\}=\{2,6,8,10,12,14,18\}$.
\item[$\lie{E}_6$:] roots $\sum_{i=1}^8 a_i \one_i$ of $\lie{E}_8$ with $a_1=a_2$ and $\sum_{i=3}^8 a_i=0$,
simple roots are those of~$\lie{E}_8$ except $\one_1-\one_2$ and $\one_2-\one_3$.
The $2^7\,3^4\,5=51840$-element Weyl group is $\operatorname{Aut}(\Lie{PSp}_4(\FF_3))$. Degrees of invariants are $\{d_i\}=\{2,5,6,8,9,12\}$.
\item[$\lie{F}_4$:] roots $\pm\one_i$, $\pm\one_i\pm\one_j$, $\frac{1}{2}(\pm\one_1\pm\one_2\pm\one_3\pm\one_4)$,
simple roots $\one_1-\one_2$, $\one_2-\one_3$, $\one_3$, $-\frac{1}{2}(\one_1+\one_2+\one_3+\one_4)$.
It has an $1152$-element Weyl group and $\{d_i\}=\{2,6,8,12\}$.
\item[$\lie{G}_2$:] $12$~roots $\rme^{2\pi\I k/6},\rme^{2\pi\I (2k+1)/12}\sqrt{3}\in\CC$ for $0\leq k< 6$, simple roots~$1$ and $\rme^{5\pi\I/6}\sqrt{3}$.
The $12$-element Weyl group is the dihedral group~$D_6$, and $\{d_i\}=\{2,6\}$.
\end{description}
The Coxeter number $h(\lie{G})=(\dim\lie{G}/\rank\lie{G})-1$ is the largest~$d_i$. A Coxeter element is the product of all simple reflections, in any order. Its eigenvalues $\rme^{2\pi\I(d_i-1)/h}$ come in conjugate pairs.
\paragraph[Real simple Lie algebras]{A real simple Lie algebra} is a complex algebra (see above) or a real form of it.
Let
$\lie{sp}(m,n)=\lie{usp}(2m,2n)=\lie{u}(m,n,\HH)$,
$\lie{su}^*(2n)=\lie{sl}(n,\HH)=\{\Re\Tr M=0\text{ in }\lie{gl}(n,\HH)\}\simeq\lie{gl}(n,\HH)/\RR$,
$\lie{so}^*(2n)=\lie{o}(n,\HH)$.
A Lie algebra is called compact if it exponentiates to a compact Lie group.
In $\lie{E}_{r(s)}$, $s$ is the number of $(\text{non-compact})-(\text{compact})$ generators.
The maximal compact subalgebra of a complex algebra is its compact real form.
\begin{tab}{@{}c@{ }l@{\,}l@{ \ }l}
& Real form & Max compact subalgebra & Range \\
\midrule
\multirow{4}{*}{\rotatebox{90}{$\lie{sl}(n,\CC)$}}
& $\lie{su}(n)$ & compact & \\
& $\lie{sl}(n,\RR)$ & $\supset\lie{so}(n)$ & \\
& $\lie{su}(n-p,p)$ & $\supset\lie{su}(n-p)\oplus \lie{su}(p)\oplus \lie{u}(1)$ & $0<p<n$ \\
& $\lie{su}^*(n)$ & $\supset\lie{usp}(n)$ & $n$ even \\
\midrule
\multirow{3}{*}{\rotatebox{90}{$\lie{so}(n,\CC)$}}
& $\lie{so}(n)$& compact & \\
& $\lie{so}(p,n-p)$& $\supset\lie{so}(p)\oplus \lie{so}(n-p)$ & $0<p<n$ \\
& $\lie{so}^*(n)$ & $\supset\lie{u}(n/2)$ & $n$ even \\
\midrule
\multirow{3}{*}{\rotatebox{90}{$\lie{sp}(2n,\CC)\!\!$}}
& $\lie{usp}(2n)$ & compact & \\
& $\lie{sp}(2n,\RR)$ & $\supset\lie{u}(n)$ & \\
& $\lie{usp}(2n-2p,2p)$ & $\supset\lie{usp}(2n-2p)\oplus \lie{usp}(2p)$ & $0<p<n$ \\
\midrule
\multicolumn{4}{c}{%
\begin{tabular}[c]{l@{ }l}
$\lie{E}_{6(-78)}$ & compact \\
$\lie{E}_{6(-26)}$ & $\supset\lie{F}_4$ \\
$\lie{E}_{6(-14)}$ & $\supset\lie{so}(10)\oplus \lie{so}(2)$\\
$\lie{E}_{6(2)}$ & $\supset\lie{su}(6)\oplus \lie{su}(2)$\\
$\lie{E}_{6(6)}$ & $\supset\lie{usp}(8)$\\
\midrule
$\lie{E}_{7(-133)}$& compact \\
$\lie{E}_{7(-25)}$& $\supset\lie{E}_6\oplus \lie{so}(2)$ \\
$\lie{E}_{7(-5)}$& $\supset\lie{so}(12)\oplus \lie{su}(2)$ \\
$\lie{E}_{7(7)}$& $\supset\lie{su}(8)$
\end{tabular}\quad
\begin{tabular}[c]{l@{\,}l}
$\lie{E}_{8(-248)}$& compact\\
$\lie{E}_{8(-24)}$&$\supset\lie{E}_7\oplus \lie{su}(2)$\\
$\lie{E}_{8(8)}$&$\supset\lie{so}(16)$\\
\midrule
$\lie{G}_{2(-14)}$ & compact \\
$\lie{G}_{2(2)}$ & $\supset\lie{su}(2)\oplus \lie{su}(2)$ \\
\midrule
$\lie{F}_{4(-52)}$ & compact \\
$\lie{F}_{4(-20)}$ & $\supset\lie{so}(9)$ \\
$\lie{F}_{4(4)}$ & $\supset\lie{usp}(6)\oplus \lie{su}(2)$ \\
\end{tabular}
}\\
\end{tab}
\paragraph{Accidental isomorphisms.}
\begin{center}
\vspace{-2.5\baselineskip}
\begin{minipage}[t]{.6\linewidth}%
\begin{align*}
\lie{so}(2)&=\lie{so}^*(2)=\lie{u}(1)\\
\lie{so}(1,1)&=\RR\\
\lie{so}(3)&= \lie{su}(2)=\lie{su}^*(2)=\lie{usp}(2)\\
\lie{so}(2,1) &=\lie{su}(1,1)=\lie{sl}(2,\RR)=\mathrlap{\lie{sp}(2,\RR)}\\
\lie{so}(4)&=\lie{su}(2)\oplus \lie{su}(2)\\
\lie{so}(3,1)&=\lie{sl}(2,\CC)=\lie{sp}(2,\CC)\\
\lie{so}(2,2)&=\lie{sl}(2,\RR)\oplus \lie{sl}(2,\RR)\\
\lie{so}^*(4)&=\lie{sl}(2,\RR)\oplus \lie{su}(2)
\end{align*}
\end{minipage}%
\begin{minipage}[t]{.4\linewidth}
\begin{align*}
\lie{so}(5)&=\lie{usp}(4)\\
\lie{so}(4,1)&=\lie{usp}(2,2)\\
\lie{so}(3,2)&=\lie{sp}(4,\RR)\\
\lie{so}(6)&=\lie{su}(4)\\
\lie{so}(5,1)&=\lie{su}^*(4)\\
\lie{so}(4,2)&=\lie{su}(2,2)\\
\lie{so}(3,3)&=\lie{sl}(4,\RR)\\
\lie{so}^*(6)&=\lie{su}(3,1)\\
\lie{so}^*(8)&=\lie{so}(6,2)
\end{align*}
\end{minipage}
\end{center}
\paragraph{ADE classification}
of symmetric matrices with eigenvalues in $(-2,2)$ and $\ZZ_{\geq 0}$ entries (adjacency matrices of ADE diagrams),
of simply laced simple Lie algebras,
of binary polyhedral groups~$\Gamma$ (discrete subgroups of $\Lie{SU}(2)$) and du~Val singularities $\CC^2/\Gamma\simeq$(zeros of Kleinian polynomial),
of integers $1\leq p\leq q\leq r$ with $1/p+1/q+1/r>1$,
of singularities with no moduli (Arnold) hence of $\Nsusy=2$ minimal models ($c<3$),
of $\Nsusy=0$ unitary minimal models ($c<1$),
of quivers of finite type,\ldots{}
\begin{tab}{*{3}{>{$}l<{$}}}\label{page:ADE-polynomial}
\lie{G} & (p,q,r) & \text{Kleinian polynomial} \\
\midrule
\lie{A}_k & (1,q,1+k-q) & w^2+x^2+y^{k+1} \\
\lie{D}_k & (2,2,k-2) & w^2+x^2y+y^{k-1} \\
\lie{E}_6 & (2,3,3) & w^2+x^3+y^4 \\
\lie{E}_7 & (2,3,4) & w^2+x^3+xy^3 \\
\lie{E}_8 & (2,3,5) & w^2+x^3+y^5 \\
\end{tab}
\subsection{Lie groups}
% TODO: say Lie subgroups can be horrible, such as irrational line in $T^2$
% TODO: Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen (1988), D. Joyce (1992) construct (all) left-invariant hypercomplex structures on (simply-connected?) compact (reductive?) Lie groups: any simple Lie groups become ``left-invariantly hypercomplex'' once multiplied by an appropriate torus.
\paragraph{Basics.} The identity component~$G_0$ is a normal subgroup: $G/G_0$ is the group of components. The maximal compact subgroup~$K$ is unique up to conjugation.
% TODO: is every compact connected Lie group reductive?
\paragraph[Compact connected Lie groups]{Every compact connected Lie group}~$K$
is a quotient of $\Lie{U}(1)^n\times\prod_{i=1}^m K_i$ by a finite subgroup~$\Gamma$ of its center,
where~$K_i$ are simple, compact, simply-connected, connected.
Then $\pi_1(K)/\ZZ^n\simeq\Gamma$ for some embedding $\ZZ^n\hookrightarrow\pi_1(K)$,
and the center of~$K$ is $Z(K)=\bigl(\Lie{U}(1)^n\times\prod_{i=1}^m Z(K_i)\bigr)/\Gamma$.
Center of all such~$K_i$:
\hspace{0pt plus 10pt}$Z(\Lie{SU}(n))=\ZZ_n$,
\hspace{0pt plus 10pt}$Z(\Lie{USp}(2n))=\ZZ_2$,
\hspace{0pt plus 10pt}$Z(\Lie{Spin}(n\geq 3))=(\ZZ_2$ for $n$ odd, $\ZZ_4$ for $n/2$ odd, $\ZZ_2^2$ otherwise),
\hspace{0pt plus 10pt}$Z(\widetilde{\Lie{E}}_{6(-78)})=\ZZ_3$,
\hspace{0pt plus 10pt}$Z(\widetilde{\Lie{E}}_{7(-133)})=\ZZ_2$,
while $\Lie{E}_{8(-248)}$, $\Lie{F}_{4(-52)}$, $\Lie{G}_{2(-14)}$ have no center.
Named quotients: $\Lie{SO}(n)=\Lie{Spin}(n)/\ZZ_2$ and $\Lie{P}G=G/Z(G)$ for $G=\Lie{SU},\Lie{USp},\Lie{SO}$ (also $\Lie{U},\Lie{GL},\Lie{SL}$).
The other two quotients $\Lie{Spin}(4n)/\ZZ_2$ have no name.
\paragraph{Real connected simple Lie groups}
are the simply-con\-nected~$\widetilde{G}$ (classified by simple Lie algebras)
and their quotients by a subgroup $\Gamma\subset Z(\widetilde{G})$ of the center;
equivalently, covers of the center-free $G_{\text{cf}}=\widetilde{G}/Z(\widetilde{G})$.
One has $\pi_1(\widetilde{G}/\Gamma)=\Gamma$ and $Z(\widetilde{G}/\Gamma)=Z(\widetilde{G})/\Gamma$.
The algebraic universal cover~$\widetilde{G}_{\text{alg}}$ (largest with a faithful finite-dimensional representation) may be a quotient of~$\widetilde{G}$.
We define $\pi_1^{\text{alg}}(\widetilde{G}_{\text{alg}}/\Gamma)=\Gamma$.
For each real simple Lie algebra~$\lie{g}$, we tabulate: $G_{\text{cf}}$~as a quotient of $\widetilde{G}_{\text{alg}}$; the (topological)~$\pi_1$; the real rank~$r_{\Re}$; and
the maximal compact subgroup $K\subset G_{\text{cf}}$.
Below, $\iota(l)=(1$ for $l$~odd, $2$ otherwise), $p+q=n$ with $p,q\geq 1$, and $2k=n$ when $n$ is even.
For $\lie{sl}(2)$ use $\Lie{SU}(2)=\Lie{Sp}(2)$, $\Lie{SL}(2,\RR)=\Lie{Sp}(2,\RR)$, $\Lie{SL}(2,\CC)=\Lie{Sp}(2,\CC)$.
\begin{tab}{@{}c@{ }llcr}
& $\widetilde{G}_{\text{alg}}/\pi_1^{\text{alg}}(G_{\text{cf}})$ & $K$ & $\pi_1$ & $r_{\Re}$ \\
\midrule
\multirow{5}{*}{\rotatebox{90}{$\lie{sl}(n\geq 3)$}}
& $\Lie{SU}(n)/\ZZ_n$ & $\Lie{SU}(n)/\ZZ_n$ & $\ZZ_n$ & $0$ \\
& $\Lie{SL}(n,\RR)/\ZZ_{\iota(n)}$ & $\Lie{PSpin}(n)$\tabfoot{1}\tabfoot{2} & \clap{$Z(\Lie{Spin}(n))$\tabfoot{1}\tabfoot{2}} & $n-1$ \\
& $\Lie{SU}(p,q)/\ZZ_{p+q}$ & $\frac{\Lie{SU}(p)\times\Lie{SU}(q)\times\Lie{U}(1)}{\ZZ_{pq/\gcd(p,q)}}$\tabfoot{3} & $\ZZ$ & $\min(p,q)$ \\
& $\Lie{SU}^*(2k)/\ZZ_2$ & $\Lie{USp}(2k)/\ZZ_2$ & $\ZZ_2$ & $k-1$ \\
& $\Lie{SL}(n,\CC)/\ZZ_n$ & $\Lie{SU}(n)/\ZZ_n$ & $\ZZ_n$ & $n-1$ \\
\midrule
\multirow{4}{*}{\rotatebox{90}{$\lie{so}(n\geq 3)$}}
& $\Lie{PSpin}(n)$\tabfoot{1} & $\Lie{PSpin}(n)$ & \clap{$Z(\Lie{Spin}(n))$\tabfoot{1}} & $0$ \\
& $\Lie{PSpin}(p,q)$\tabfoot{1} & \rlap{$\frac{\Lie{SO}(p){\times}\Lie{SO}(q)}{\ZZ_2\text{ if $p$, $q$ even}}$} & $\Gamma$\tabfoot{4} & $\min(p,q)$ \\
& $\Lie{SO}^*(2k)/\ZZ_2$ & $\Lie{U}(k)/\ZZ_2$ & \clap{$\ZZ_{\iota(k)}\times\ZZ$} & $\lfloor k/2\rfloor$ \\
& $\Lie{PSpin}(n,\CC)$ & $\Lie{PSpin}(n)$ & \clap{$Z(\Lie{Spin}(n))$\tabfoot{1}} & $\lfloor n/2\rfloor$ \\
\midrule
\multirow{4}{*}{\rotatebox{90}{$\lie{sp}(2n\geq 2)$}}
& $\Lie{USp}(2n)/\ZZ_2$ & $\Lie{USp}(2n)/\ZZ_2$ & $\ZZ_2$ & $0$ \\
& $\Lie{Sp}(2n,\RR)/\ZZ_2$ & $\Lie{U}(n)/\ZZ_2$ & \clap{$\ZZ_{\iota(n)}\times\ZZ$} & $n$ \\
& $\Lie{USp}(2p,2q)/\ZZ_2$ & $\frac{\Lie{USp}(2p)\times \Lie{USp}(2q)}{\ZZ_2}$ & $\ZZ_2$ & $\min(p,q)$ \\
& $\Lie{Sp}(2n,\CC)/\ZZ_2$ & $\Lie{USp}(2n)/\ZZ_2$ & $\ZZ_2$ & $n$ \\
\end{tab}
\tabfoot{1}
For $r+s\geq 3$,
$\Lie{PSpin}(r,s)=\Lie{Spin}(r,s)/Z(\Lie{Spin}(r,s))$ and
$Z(\Lie{Spin}(r,s))=(\ZZ_2$ if $r$ or $s$ odd, $\ZZ_4$ if $\frac{r+s}{2}$ odd, else~$\ZZ_2^2$).
\tabfoot{2}
Exception: for $n=2$, $K=\Lie{SO}(2)/\ZZ_2$ and $\pi_1=\ZZ$.
\tabfoot{3}
$K\ni\overline{(A,B,\lambda)}\mapsto\setlength{\arraycolsep}{0pt}\begin{pmatrix}\lambda^{q/(p+q)}A&0\\0&\lambda^{-p/(p+q)}B\end{pmatrix}\in\Lie{PSU}(p,q)$.
\tabfoot{4}
$\Gamma=\pi_1(\Lie{SO}(p))\times\pi_1(\Lie{SO}(q))$ for $p$ or $q$ odd (each factor is~$\ZZ_2$ except $\pi_1(\Lie{SO}(1))=0$ and $\pi_1(\Lie{SO}(2))=\ZZ$);
otherwise $\Gamma\subset\pi_1(\Lie{SO}(p)/\ZZ_2)\times\pi_1(\Lie{SO}(q)/\ZZ_2)$ consists of $(\gamma_p,\gamma_q)$ such that both or neither~$\gamma$ is in the corresponding $\pi_1(\Lie{SO})\subset\pi_1(\Lie{SO}/\ZZ_2)$.
\begin{tab}{@{}c@{ }llcr}
& $\widetilde{G}_{\text{alg}}/\pi_1^{\text{alg}}(G_{\text{cf}})$ & $K$ & $\pi_1$ & $r_{\Re}$ \\
\midrule
& $\widetilde{\Lie{E}}_{6(-78)}/\ZZ_3$ & $=\Lie{E}_{6(-78)}$ & $\ZZ_3$ & $0$ \\
& $\Lie{E}_{6(-26)}$ & $\Lie{F}_{4(-52)}$ & $1$ & $2$ \\
& $\widetilde{\Lie{E}}_{6(-14)}/\ZZ$ & $\Lie{Spin}(10)\times\Lie{U}(1)/?$ & $\ZZ$ & $2$ \\
& $\widetilde{\Lie{E}}_{6(2)}/\ZZ_6$ & $(\Lie{SU}(6)/\ZZ_6)\times\Lie{SU}(2)$ & $\ZZ_6$ & $4$ \\
& $\widetilde{\Lie{E}}_{6(6)}/\ZZ_2$ & $\Lie{USp}(8)/\ZZ_2$ & $\ZZ_2$ & $6$ \\
& $\widetilde{\Lie{E}}^{\CC}_{6}/\ZZ_3$ & $\Lie{E}_{6(-78)}$ & $\ZZ_3$ & $6$ \\
\midrule
& $\widetilde{\Lie{E}}_{7(-133)}/\ZZ_2$ & $=\Lie{E}_{7(-133)}$ & $\ZZ_2$ & $0$ \\
& $\widetilde{\Lie{E}}_{7(-25)}/\ZZ$ & $\Lie{E}_{6(-78)}\times\Lie{U}(1)/?$ & $\ZZ$ & $3$ \\
& $\widetilde{\Lie{E}}_{7(-5)}/\ZZ_2^2$ & $\Lie{Spin}(12)\times\Lie{SU}(2)/\ZZ_2^2$ & $\ZZ_2^2$ & $4$ \\
& $\widetilde{\Lie{E}}_{7(7)}/\ZZ_4$ & $\Lie{SU}(8)/\ZZ_4$ & $\ZZ_4$ & $7$ \\
& $\widetilde{\Lie{E}}^{\CC}_{7}/\ZZ_2$ & $\Lie{E}_{7(-133)}$ & $\ZZ_2$ & $7$ \\
\midrule
& $\Lie{E}_{8(-248)}$ & $\Lie{E}_{8(-248)}$ & $1$ & $0$ \\
& $\widetilde{\Lie{E}}_{8(-24)}/\ZZ_2$ & $\widetilde{\Lie{E}}_{7(-133)}\times\Lie{SU}(2)/\ZZ_2$ & $\ZZ_2$ & $4$ \\
& $\widetilde{\Lie{E}}_{8(8)}/\ZZ_2$ & $\Lie{SO}(16)/\ZZ_2$ & $\ZZ_2$ & $8$ \\
& $\Lie{E}^{\CC}_{8}$ & $\Lie{E}_{8(-248)}$ & $1$ & $8$ \\
\midrule
& $\Lie{F}_{4(-52)}$ & $\Lie{F}_{4(-52)}$ & $1$ & $0$ \\
& $\widetilde{\Lie{F}}_{4(-20)}/\ZZ_2$ & $\Lie{Spin}(9)/\ZZ_2$ & $\ZZ_2$ & $1$ \\
& $\widetilde{\Lie{F}}_{4(4)}$ & $\Lie{USp}(6)\times\Lie{SU}(2)/\ZZ_2$ & $\ZZ_2$ & $4$ \\
& $\Lie{F}^{\CC}_{4}$ & $\Lie{F}_{4(-52)}$ & $1$ & $4$ \\
\midrule
& $\Lie{G}_{2(-14)}$ & $\Lie{G}_{2(-14)}$ & $1$ & $0$ \\
& $\widetilde{\Lie{G}}_{2(2)}/\ZZ_2$ & $\Lie{SU}(2)\times\Lie{SU}(2)/\ZZ_2$ & $\ZZ_2$ & $4$ \\
& $\Lie{G}^{\CC}_{2}$ & $\Lie{G}_{2(-14)}$ & $1$ & $4$ \\
\end{tab}
\qquad\llap{\rotatebox{90}{\mbox{}\rlap{\qquad\qquad\textbf{Discrete groups in this table should not be trusted.}}}}
% TODO: outer automorphism groups
% \paragraph{Classical Lie groups}
% TODO: give a construction of all real forms of classical Lie groups
% TODO: Check we used consistently the name "\Lie{USp}(2n)=\Lie{Sp}(n)" for the compact symplectic group of rank n.
\paragraph{Spin and Pin groups.}
$\Lie{SO}(n)$ has a double cover $\Lie{Spin}(n)$.
Since $\pi_0(\Lie{O}(n))=\ZZ_2$ there are two double covers:
$\Lie{Pin}_+(n)$ in which a reflection~$R$ obeys~$R^2=1$,
and $\Lie{Pin}_-(n)$ in which $R^2=(-1)^F$.
For $p,q\geq 1$, $\pi_0(\Lie{O}(p,q))=\pi_0(\Lie{O}(p))\times\pi_0(\Lie{O}(q))=\ZZ_2^2$;
the identity component $\Lie{SO}_+(p,q)$ has a double cover $\Lie{Spin}(p,q)$.
The eight double covers of $\Lie{O}(p,q)$ differ in whether $R^2$, $T^2$ and $(RT)^2$ are $+1$ or $(-1)^F$.
\paragraph{Accidental isomorphisms} (real reductive Lie groups)
$\RR/\ZZ=\Lie{U}(1)$; $\Lie{SU}(2)=\Lie{Spin}(3)\twoheadrightarrow\Lie{SO}(3)$; \ldots{}
% TODO: complete list of isomorphisms, perhaps describe embeddings e.g. of SO(1,3)=SL(2,C) in Spin(4,C)=SL(2,C)xSL(2,C)?
\paragraph{Homotopy.} Any connected Lie group is homeomorphic to its maximal compact subgroup~$K$ times a Euclidean space~$\RR^p$.
All $\pi_{j\geq 1}(K)$ are abelian and finitely generated, $\pi_2(K)=0$, $\pi_3(K)=\ZZ^m$ where $m$~counts simple factors in a finite cover $\Lie{U}(1)^n\times\prod_{i=1}^m K_i\twoheadrightarrow K$, and $\pi_j(K)=\prod_{i=1}^m\pi_j(K_i)$ for $j\geq 2$.
For any~$G$ there exists $\prod_{i=1}^{\rank G} S^{2d_i-1}\to G$ which induces isomorphisms of rational (\ie, torsion-free part of) homotopy/\allowbreak cohomology groups where $d_i$~are the degrees of fundamental invariants. For compact simple~$K$,
\begin{center}
\vspace{-1.3\baselineskip}
\begin{minipage}[t]{.5\linewidth}%
\begin{tab}{@{}>{$}l<{$}@{ }>{$}l<{$}@{}}
\text{Group} & (2d_i-1) \\\midrule
\Lie{A}_n & 3,5,\ldots,2n+1\\
\Lie{B}_n,\Lie{C}_n & 3,7,\ldots,4n-1\\
\Lie{D}_n & 3,7,\ldots,4n-5,2n-1\\
\end{tab}
\end{minipage}%
\begin{minipage}[t]{.5\linewidth}
\begin{tab}{@{}>{$}l<{$}@{ }>{$}l<{$}@{}}
\Lie{E}_6 & 3,9,11,15,17,23\\
\Lie{E}_7 & 3,11,15,19,23,27,35\\
\Lie{E}_8 & 3,15,23,27,35,39,47,59\\
\Lie{F}_4 & 3,11,15,23\\
\Lie{G}_2 & 3,11\\
\end{tab}
\end{minipage}
\end{center}
\vspace{-.6\baselineskip}
$\pi_{j\geq 2}(G)$ has a factor~$\ZZ$ for each~$S^j$ above, and some torsion.
Explicitly, $\pi_j(\Lie{SU}(n))$ is $\ZZ$~for odd~$j<2n$, $0$~for even~$j<2n$, and is pure torsion for $j\geq 2n$.
Similarly, $\pi_{j<4n+2}(\Lie{USp}(2n))$ is $\ZZ$~for $j\equiv 3,7\bmod{8}$, $\ZZ_2$~for $j\equiv 4,5\bmod{8}$, and $0$~otherwise.
% http://www.maths.ed.ac.uk/~aar/surgery/uicc/mimura.pdf claims the table is the "well-known Hopf theorem"
% See Abanov's http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cpA1-upload.pdf
% TODO: say somewhere $\Lie{SU}(2)$ is homeomorphic to $S^3$.
\subsection{Simple Lie superalgebras}
\paragraph{Classical Lie superalgebras:}
the bosonic algebra acts on the fermionic generators in a completely reducible representation.
This excludes Cartan-type superalgebras $\lie{W}(n)$, $\lie{S}(n)$, $\lie{\tilde S}(n)$ and $\lie{H}(n)$.
In this table, $m,n\geq 1$ and we do not list purely bosonic Lie algebras.
The factor $\CC$ of $\lie{sl}(m|n)$ must be removed if $m=n$.
\begin{tab}{lll}
& Bosonic algebra & Fermionic repr. \\\midrule
$\lie{sl}(m|n,\CC)$ & $\lie{sl}(m,\CC)\oplus \lie{sl}(n,\CC)\oplus\CC$ & $(m,\overline{n})\oplus(\overline{m},n)$ \\
$\lie{osp}(m|2n,\CC)$ & $\lie{so}(m,\CC) \oplus \lie{sp}(2n,\CC)$ & $(m,2n)$ \\
$\lie{D}(2,1,\alpha,\CC)$ & $\lie{sl}(2,\CC)^3$ & $(2,2,2)$ \\
$\lie{F}(4,\CC)$ & $\lie{so}(7,\CC)\oplus \lie{sl}(2,\CC)$ & $(8,2)$ \\
$\lie{G}(3,\CC)$ & $\lie{G}_2\oplus \lie{sl}(2,\CC)$ & $(7,2)$ \\
$\lie{P}(m,\CC)$ & $\lie{sl}(m+1,\CC)$ & $\text{sym}\oplus\overline{\text{antisym}}$ \\
$\lie{Q}(m,\CC)$ & $\lie{sl}(m+1,\CC)$ & adjoint\\
\end{tab}
\paragraph{Real forms of Lie superalgebras,}
starting from their compact form ($p=q=0$). $\lie{P}(m)$ has no compact form.
Here, $m,n\geq 1$, $0\leq p\leq m/2$, $0\leq q\leq n/2$.
The forms $\lie{su}^*$, $\lie{osp}^*$, $\lie{Q}^*$ only exist for even rank; $\lie{sl'}$ only if $m=n$.
\begin{tab}{*{2}{>{$}l<{$}}}
\text{Real form} & \text{Bosonic algebra} \\ \midrule
\lie{su}(m-p,p|n-q,q) & \lie{su}(m-p,p)\oplus \lie{su}(n-q,q)\oplus \lie{u}(1)\tabfoot{1}\\
\lie{sl}(m|n) & \lie{sl}(m,\RR)\oplus \lie{sl}(n,\RR)\oplus \lie{so}(1,1)\tabfoot{1} \\
\lie{sl'}(n|n) \quad\, (m=n)& \lie{sl}(n,\CC)\\
\lie{su}^*(m|n) \:\: (m,n \text{ even}) & \lie{su}^*(m)\oplus \lie{su}^*(n)\oplus \lie{so}(1,1)\tabfoot{1}\\
\midrule
\lie{osp}(m-p,p|2n) & \lie{so}(m-p,p)\oplus \lie{sp}(2n,\RR) \\
\multicolumn{2}{l}{$\lie{osp}^*(m|2n-2q,2q)$ ($m$ even)\quad $\lie{so}^*(m)\oplus \lie{usp}(2n-2q,2q)$} \\
\midrule
\lie{D}^p(2,1,\alpha) \;\tabfoot{2} & \lie{so}(4-p,p)\oplus \lie{sl}(2,\RR)\ (p=0,1,2)\\
\midrule
\lie{F}^p(4) \text{ for $p=0,3$} & \lie{so}(7-p,p)\oplus \lie{sl}(2,\RR) \\
\lie{F}^p(4) \text{ for $p=1,2$} & \lie{so}(7-p,p)\oplus \lie{su}(2) \\
\midrule
\lie{G}_s(3) \text{ for $s=-14,2$} & \lie{G}_{2(s)}\oplus \lie{sl}(2,\RR) \\
\midrule
\lie{P}(m) & \lie{sl}(m+1,\RR) \\
\midrule
\lie{UQ}(m-p,p) & \lie{su}(m+1-p,p) \\
\lie{Q}(m) & \lie{sl}(m+1,\RR) \\
\lie{Q}^*(m) \quad (m \text{ odd}) & \lie{su}^*(m+1) \\
\end{tab}
\tabfoot{1}
For $m=n$, omitting $\lie{u}(1)$ and $\lie{so}(1,1)$ factors defines $\lie{psu}$, $\lie{psl}$, $\lie{psu}^*$.
One can also project down to a single bosonic factor.
\tabfoot{2}
The three $\lie{sl}(2)$ bosonic factors of $\lie{D}(2,1,\alpha)$ appear with weights $1$, $\alpha$ and $-1-\alpha$ in fermion anticommutators.
For $\lie{D}^0$ and $\lie{D}^2$, $\alpha$ is real. For $\lie{D}^1$, $\alpha=1+ia$ with $a$ real.
\smallskip
\paragraph{Accidental isomorphisms.} Incomplete, may have errors.
Exchanging $m\leftrightarrow n$ in $\lie{sl}(m|n,\CC)$, or the two sides of ``$|$'' in its real forms, or $p\leftrightarrow q$ or $r\leftrightarrow s$ in $\lie{su}(p,q|r,s)$ or $\lie{osp}(p,q|2n)$ or $\lie{osp}^*(2k|2p,2q)$ is an isomorphism.
Exception: $\lie{UQ}(p-1,q)=\lie{UQ}(q-1,p)$.
\begin{center}
\vspace{-2.3\baselineskip}
\begin{minipage}[t]{.45\linewidth}%
\begin{align*}
\lie{osp}(2|2,\CC)&=\lie{sl}(2|1,\CC)\\
\lie{osp}^*(2|2)&=\lie{su}(2|1)\\
\lie{osp}(2|2)&=\lie{su}(1,1|1)\\
\lie{osp}(1,1|2)&=\lie{sl}(2|1)\\
\lie{sl}(2|2)&=\lie{su}(1,1|1,1)\\
\lie{su}^*(2|2)&=\lie{su}(2|2)
\end{align*}
\end{minipage}%
\begin{minipage}[t]{.55\linewidth}
\begin{align*}
\lie{D}(2,1,\alpha=1,\CC)&=\lie{osp}(4|2,\CC)\\
\lie{D}^p(2,1,\alpha=1)&=\lie{osp}(4-p,p|2)\\
\lie{UQ}(1,0)&=\lie{Q}^*(1)\\
\lie{UQ}(0,1)&=\lie{Q}(1)
\end{align*}
\end{minipage}
\end{center}
% TODO: complete this list
\subsection{Lie supergroups}
\subsection{Representations}
% TODO: representations of complex Lie algebras
% TODO: representations of real Lie algebras
\paragraph{Complex dimensions of smalest irreps}
\begin{tab}{ll}
$\Lie{E}_6$ & \href{https://oeis.org/A121737}{$1,27,27,78,351,351,351,351,650,1728,1728,\dots$} \\
$\Lie{E}_7$ & \href{https://oeis.org/A121736}{$1,56,133,912,1463,1539,6480,7371,8645,24320,\dots$} \\
$\Lie{E}_8$ & \href{https://oeis.org/A121735}{$1,248,3875,27000,30380,147250,779247,1763125,\dots\!$} \\
$\Lie{F}_4$ & \href{https://oeis.org/A121738}{$1,26,52,273,324,1053,1053,1274,2652,4096,\dots$} \\
$\Lie{G}_2$ & \href{https://oeis.org/A104599}{$1,7,14,27,64,77,77,182,189,273,286,378,448,\dots$} \\
\end{tab}
\section{Spinors, supersymmetry}
\subsection[Spinors]{Spinors \normalsize\textnormal{(e.g.\@ \hepth{9910030})}}
\paragraph{Clifford algebra.} Let $h_{ab}$ be diagonal with $s$~`$+1$' and $t$~`$-1$', and $d=s+t$. The Clifford algebra $\{\Gamma_a,\Gamma_b\}=2h_{ab}$ has real dimension~$2^d$ and is isomorphic to a matrix algebra $M_{2^{\#}}(\bullet)$ with
\begin{tab}{>{$}r<{$}*{8}{>{$}c<{$}}}
s-t \bmod{8} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\bullet \text{\quad is} & \RR& \RR\oplus\RR & \RR & \CC & \HH & \HH\oplus\HH & \HH &\CC\\
\end{tab}
%with $\bullet = [\RR, \RR\oplus\RR, \RR, \CC, \HH, \HH\oplus\HH, \HH, \CC][s-t\bmod{8}]$.
\paragraph{Charge conjugation.} $(-\eta)\Gamma_a^T=\mathcal{C}\Gamma_a\mathcal{C}^{-1}$ are conjugate for $\eta=\pm 1$ because they obey the same algebra. Get $\mathcal{C}^T=-\varepsilon \mathcal{C}$ with $\varepsilon=\pm 1$ by transposing twice.
Let $\Gamma^{(n)}=\Gamma_{a_1\ldots a_n}$.
Using $\left(\mathcal{C}\Gamma^{(n)}\right) ^T= -\epsilon(-)^{n(n-1)/2}(-\eta)^n \mathcal{C}\Gamma^{(n)}$ find which $n\bmod{4}$ give symmetric $\mathcal{C}\Gamma^{(n)}$. The sum of $\binom{d}{n}$ must be $2^{\lfloor d/2\rfloor} (2^{\lfloor d/2\rfloor} + 1) / 2$. This fixes $\epsilon, \eta$. Odd $d$ require $\eta=(-1)^{d(d+1)/2}$ to preserve $\Gamma^{(d)}$. Even~$d$ allow two choices of signs: consult the rows $d\pm 1$.
\begin{tab}{rccc}
$d\bmod 8$ & $n$ & $\epsilon$ & $\eta$ \\\midrule
0---\llap{\multirow{2}{*}{2$\langle$}}1 & 0, 1 & $-1$ & $-1$\\
\multirow{2}{*}{4$\langle$}3 & 1, 2 & $+1$ & $+1$\\
\multirow{2}{*}{6$\langle$}5 & 2, 3 & $+1$ & $-1$\\
0---7 & 0, 3 & $-1$ & $+1$\\
\end{tab}
\paragraph{Reduced spinors.}
$M_{ab}\in \lie{so}(s,t)$ acts as $\gamma_a \gamma_b$ on representations of the Clifford algebra.
But the $2^{\lceil d/2\rceil}$-dimensional representation is not irreducible as a representation of $\lie{so}(s,t)$.
In even~$d$, Weyl (or chiral) spinors $\Gamma^{(d)}\lambda=\pm\lambda$ have $2^{d/2-1}$ real components.
Let $B$~be defined by $\Gamma_a^*=-\eta(-1)^t B\Gamma_a B^{-1}$.
Majorana spinors $\lambda^*=B\lambda$ exist for $s-t\equiv 0,\pm 1,\pm 2\bmod{8}$;
the case $s-t\equiv\pm 2$ requires $\eta=\mp (-1)^{d/2}$.
When $s-t\equiv 3,4,5$, a set of $2n$ spinors can be symplectic Majorana: $(\lambda^I)^*=B\Omega_{IJ}\lambda^J$ for $\Omega=((0,\one_n);(-\one_n,0))$.
(Symplectic) Majorana--Weyl spinors exist for $s-t\equiv 0,4\bmod{8}$.
The table also includes the real dimension of the minimal spinor.
\begin{tab}{c@{ }>{\footnotesize}l@{}>{\footnotesize}r@{}*{4}{>{ }l@{ }r<{ }}}
d & & &\multicolumn{2}{c}{$t\equiv 0$} &\multicolumn{2}{c}{$1$}
& \multicolumn{2}{c}{$2$} & \multicolumn{2}{c}{$3\bmod{4}$} \\ \midrule
1 & (D&2) & M & 1 & M & 1 & & & & \\
2 & (W&2) & M$^-$ & 2 & MW & 1 & M$^+$ & 2 & & \\
3 & (D&4) & s & 4 & M & 2 & M & 2 & s & 4 \\
4 & (W&4) & sW & 4 & M$^+$& 4 & MW & 2 & M$^-$ & 4 \\
5 & (D&8) & s & 8 & s & 8 & M & 4 & M & 4 \\
6 & (W&8) & M$^+$ & 8 & sW & 8 & M$^-$ & 8 & MW & 4 \\
7 & (D&16) & M & 8 & s & 16 & s & 16 & M & 8 \\
8 & (W&16) & MW & 8 & M$^-$& 16 & sW & 16 & M$^+$& 16 \\
9 & (D&32) & M & 16& M & 16 & s & 32 & s & 32 \\
10& (W&32) & M$^-$ & 32& MW & 16 & M$^+$& 32 & sW & 32 \\
11& (D&64) & s & 64& M & 32 & M & 32 & s & 64 \\
12& (W&64) & sW & 64& M$^+$& 64 & MW & 32 & M$^-$ & 64\\
\end{tab}
\paragraph{Flavour symmetries} of $N$ minimal spinors.
This is also the R-symmetry of the $N$-extended superalgebra.
For (symplectic) Majorana Weyl spinors, specify $N=(N_L,N_R)$ left/right-handed.
\begin{tab}{l@{}>{$}l<{$}}
M & \hspace{-2pt}\begin{cases} \lie{u}(N)&\text{if $d$ even}\\\lie{so}(N)&\text{if $d$ odd}\end{cases}\\
MW & : \lie{so}(N_L)\times \lie{so}(N_R) \\
s & : \lie{usp}(2N) \\
sW & : \lie{usp}(2N_L)\times \lie{usp}(2N_R)\\
\end{tab}
\Eg, Lorentzian 6d $(2,0)$ has $\lie{usp}(4)\times\lie{usp}(0)$ R-symmetry.
\paragraph{Products of spinor representations.}
For odd $d=2m+1$, let $\mathcal{S}$ be a spinor representation of complex dimension $2^{m}$.
The symmetric product $S^2\mathcal{S}$ consists of $k$-forms with $k\equiv m\bmod{4}$.
Since $k$-forms and $(d-k)$-forms are the same representation, other descriptions can be given.
For the antisymmetric product $\Lambda^2\mathcal{S}$, take $k\equiv m-1\bmod{4}$.
See the list of forms in the table.
\begin{tab}{>{$}l<{$}*{6}{>{$}l<{$}}}
d & 1 & 3 & 5 & 7 & 9 & 11\\
\dim_{\CC}\mathcal{S} & 1 & 2 & 4 & 8 & 16 & 32\\\midrule
S^2\mathcal{S} & 0 & 1 & 2 & 0,3 & 0,1,4 & 1,2,5\\
\Lambda^2\mathcal{S} & . & 0 & 0,1 & 1,2 & 2,3 & 0,3,4\\
\end{tab}
For even $d=2m$, let $\mathcal{S}_{\pm}$ be the Weyl spinor representations of complex dimension $2^{m-1}$.
Then
\begin{center}
\vspace{-.6\baselineskip}
$\begin{aligned}[t]
S^2(\mathcal{S}_{+}\oplus\mathcal{S}_{-})&=S^2\mathcal{S}_{+}\oplus(\mathcal{S}_{+}\otimes\mathcal{S}_{-})\oplus S^2\mathcal{S}_{-}\\
\Lambda^2(\mathcal{S}_{+}\oplus\mathcal{S}_{-})&=\Lambda^2\mathcal{S}_{+}\oplus(\mathcal{S}_{+}\otimes\mathcal{S}_{-})\oplus \Lambda^2\mathcal{S}_{-}
\end{aligned}$
\end{center}
\vspace{-.6\baselineskip}
The tensor product $\mathcal{S}_{+}\otimes\mathcal{S}_{-}$ consists of $(m-1-2j)$-forms for $0\leq j\leq (m-1)/2$.
The symmetric products $S^2\mathcal{S}_{\pm}$ decompose into the (anti)-self-dual $m$-forms (denoted $m^{\dagger}$) and $(m-4j)$-forms for $0<j\leq m/4$.
The antisymmetric products $\Lambda^2\mathcal{S}_{\pm}$ decompose into $(m-2-4j)$-forms for $0\leq j\leq (m-2)/4$.
\begin{tab}{>{$}l<{$}*{6}{>{$}l<{$}}}
d & 2 & 4 & 6 & 8 & 10 & 12\\
\dim_{\CC}\mathcal{S}_{\pm} & 1 & 2 & 4 & 8 & 16 & 32\\\midrule
S^2\mathcal{S}_{\pm} & 1^{\dagger} & 2^{\dagger} & 3^{\dagger} & 0,4^{\dagger} & 1,5^{\dagger} & 2,6^{\dagger}\\
\Lambda^2\mathcal{S}_{\pm} & . & 0 & 1 & 2 & 3 & 0,4\\
\mathcal{S}_{+}\otimes\mathcal{S}_{-} & 0 & 1 & 0,2 & 1,3 & 0,2,4 & 1,3,5\\
\end{tab}
\subsection{Generalities}
\paragraph{The Poincar\'e algebra} is $\RR^{s,t} \rtimes \lie{so}(s,t)$, the semi-direct product of translations by rotations.
Namely, $[P_{a},P_{b}]=0$, $[M_{ab},P_{c}]=2ih_{c[a}P_{b]}$, and $[M_{ab},M^{cd}]=4ih_{[a}^{[c}M_{b]}^{d]}$.
\smallskip
\paragraph{Super-Poincar\'e algebra.}
Add supercharges in some spinor representation~$Q$ of the Poincar\'e algebra (so $[P_{a},Q]=0$).
Their anticommutator transforms in the representation $S^2 Q$ and should include the one-form~$P$.
Depending on $s,t$ they can include other $k$-forms~$Z$, called central charges because $[P,Z]=[Z,Z]=0$.
The super-Poincar\'e algebra is $((\RR^{s,t}\times Z).Q)\rtimes (\lie{so}(s,t)\times R)$, where the R-symmetry acts on~$Q$.
This Lie superalgebra is graded: $\operatorname{gr}(\RR^{s,t}\times Z)=-2$, $\operatorname{gr}(Q)=-1$, and $\operatorname{gr}(\lie{so}(s,t)\times R)=0$.
The supertranslations consist of $(\RR^{s,t}\times Z).Q$.
\smallskip
\paragraph{Example: M-theory algebra.} $d=10+1$ super-Poincar\'e algebra with $Q=\text{Majorana}$.
Since $S^2 Q$ has $1$, $2$, and $5$-forms, there are $2$-form and $5$-form central charges $Z_{(2)}$ and~$Z_{(5)}$
(under which M2 and M5 branes are charged):
\vspace{-.5\baselineskip}
\begin{align*}
\{Q_{\alpha},Q_{\beta}\}&=(\gamma^{M}C)_{\alpha\beta} P_{M}+\frac{1}{2}(\gamma_{MN}C)_{\alpha\beta} Z_{(2)}^{MN}\\[-.5\baselineskip]
& \quad + \frac{1}{5!}(\gamma_{MNPQR}C)_{\alpha\beta} Z_{(5)}^{MNPQR}
\end{align*}
\vspace{-1\baselineskip}
\noindent Altogether the M-theory algebra is $\lie{osp}(1|32)$.
\smallskip
\paragraph{Lorentzian superconformal algebras} are the same as super $AdS_{d+1}$.
The bosonic part is $\lie{so}(d,2)$ and R-symmetries.
As a supermatrix: $\begin{pmatrix}\lie{so}(d,2)& Q+S\\ Q-S&R\end{pmatrix}$ or $\begin{pmatrix}R& Q+S\\ Q-S&\lie{so}(d,2)\end{pmatrix}$. Note that $\{Q,S\}$ contains~$R$.
For $d=2$, the finite conformal algebra is $\lie{so}(2,2)=\lie{so}(2,1)\oplus \lie{so}(2,1)$, sum of two $d=1$ algebras, so the superalgebra is sum of two $d=1$ superalgebras.
\begin{tab}{lllc}
$d$& Superalgebra& R-symm (compact) & \#Q+\#S\\ \midrule
$1$& $\lie{osp}(N|2)$ & $\lie{o}(N)$ & $2N$ \\
& $\lie{su}(N|1,1)$ &$\lie{su}(N)\oplus \lie{u}(1)$ for $N\neq 2$ &$ 4N$ \\
& $\lie{su}(2|1,1) $ &$\lie{su}(2) $ &$ 8 $\\
& $\lie{osp}(4^*|2N) $ &$\lie{su}(2)\oplus \lie{usp}(2N)$ &$ 8N $\\
& $\lie{G}_{-14}(3) $ &$\lie{G}_{2(-14)} $ &$ 14 $\\
& $\lie{F}^0(4) $ &$\lie{so}(7) $ &$ 16 $\\
& $\lie{D}^0(2,1,\alpha) $ &$\lie{su}(2)\oplus \lie{su}(2) $ &$ 8 $\\ \midrule
$3$&$ \lie{osp}(N|4) $ &$ \lie{so}(N) $ &$ 4N $\\ \midrule
$4$&$ \lie{su}(2,2|N) $ &$\lie{su}(N)\oplus \lie{u}(1)$ for $N\neq 4$&$ 8N$\\
&$ \lie{su}(2,2|4) $ &$ \lie{su}(4)$ & 32\\ \midrule
$5$&$ \lie{F}^2(4) $ &$ \lie{su}(2) $ & 16 \\ \midrule
$6$&$ \lie{osp}(8^*|N) $ &$ \lie{usp}(N)\ \ (N $ even)& $8N$ \\
\end{tab}
\paragraph{Dimensional reduction} of Lorentzian supersymmetry algebras.
The 1d column gives the number of real supercharges.
\begin{tab}{ccccccc}
10d & 6d & 5d & 4d & 3d & 2d & 1d \\\midrule
$\Nsusy=(1,0)$ & $(1,1)$ & $2$ & $4$ & $8$ & $(8,8)$ & $16$ \\
& $(1,0)$ & $1$ & $2$ & $4$ & $(4,4)$ & $8$ \\
& & & $1$ & $2$ & $(2,2)$ & $4$ \\
\end{tab}
% TODO: Euclidean case
\paragraph{Supersymmetry on symmetric curved spaces}
4d $\Nsusy=2$ supersymmetry on~$S^4$ is $\lie{osp}(2|4)$.
2d $\Nsusy=(2,2)$ supersymmetry on~$S^2$ is $\lie{osp}(2|2)$.
\subsection[Explicit susy algebras]{Explicit supersymmetry algebras}
\paragraph{4d \(\Nsusy=2\).}
$\{Q_\alpha^A,\overline{Q}_{\dot{\alpha}}^B\} =
\epsilon^{AB}P_{\alpha\dot{\alpha}}$;
$0 = \{Q_\alpha^A,Q_\beta^B\} =
\{\overline{Q}_{\dot{\alpha}}^A,\overline{Q}_{\dot{\beta}}^B\}$.
\paragraph{3d \(\Nsusy=2\).}
$\{Q_\alpha,\overline{Q}_{\beta}\} =
2\sigma^\mu_{\alpha\beta}P_\mu+2i\epsilon_{\alpha\beta}Z$ with $Z=P_3$ a central charge;
$0 = \{Q_\alpha,Q_\beta\} =
\{\overline{Q}_{\alpha},\overline{Q}_{\beta}\}$.
\subsection{Spin \(\leq 1\) supermultiplets}
Here ``$n$-susy'' means the algebra with $n$ supercharges.
See in parentheses how multiplets decompose into $n/2$-susy multiplets.
\paragraph{\(16\)-susy multiplets:}\\
vector ($8$-susy vector multiplet and adjoint hypermultiplet).
\paragraph{\(8\)-susy multiplets:}\\
vector multiplet ($4$-susy vector and adjoint chiral multiplets),\linebreak
hypermultiplet (two conjugate $4$-susy chiral multiplets)\linebreak
and half-hypermultiplet (self-conjugate $4$-susy chiral multiplet);\linebreak
in 3d and lower also twisted vector multiplet and twisted hypermultiplet.
% TODO: 4d N=2 chiral superfield $\Phi$ then $\int\dd{^4\thetacoordinate}F(\Phi)$
\paragraph{\(4\)-susy:}
vector ($V=V^\dagger$)
and chiral ($\overline{D}_{\dot{\alpha}}X=0$) multiplets;
in 3d $\Nsusy=2$ also linear multiplets ($\epsilon^{\alpha\beta} D_\alpha D_\beta\Sigma=0=\epsilon^{\alpha\beta}\overline{D}_\alpha\overline{D}_\beta\Sigma$) with lowest component a real scalar;
in 2d $\Nsusy=(2,2)$ also twisted vector, twisted chirals, semichirals, \ldots{}
\paragraph{\(2\)-susy:}
in 3d $\Nsusy=1$ and 2d $\Nsusy=(1,1)$, vector and real;
in 2d $\Nsusy=(2,0)$, vector, chiral, Fermi;
also linear,~\ldots{}
\subsection{Some other supermultiplets}
6d $\Nsusy=(2,0)$ tensor multiplet with self-dual two-form gauge field~$B$ (namely $\dd{B}=\star\dd{B}$), four spinors, five scalars.
6d $\Nsusy=(1,0)$ tensor multiplet (contains one scalar), reduces to 4d $\Nsusy=2$ vector.
6d $\Nsusy=(1,0)$ supergravity multiplet, reduces to 4d $\Nsusy=2$ supergravity multiplet and two vectors.
\paragraph{4d \(\Nsusy=1\) supercurrent multiplet}
contains stress tensor and/or R-symmetry current; is a source for supergravity.
Ferrara--Zumino supercurrent $\overline{D}^{\dot{\alpha}}J_{\alpha\dot{\alpha}}=D_\alpha X$ with $\overline{D}_{\dot{\alpha}}X=0$ contains stress tensor; sources old minimal supergravity.
R-symmetry multiplet $\overline{D}^{\dot{\alpha}}R_{\alpha\dot{\alpha}}=\chi_\alpha$,
$\overline{D}_{\dot{\alpha}}\chi_\alpha=0$, $D^\alpha\chi_\alpha=\overline{D}_{\dot{\alpha}}\overline{\chi}^{\dot{\alpha}}$
contains (conserved) R-symmetry current; sources new minimal supergravity.
Komargodski--Seiberg multiplet \arxiv{1002.2228}
$\overline{D}^{\dot{\alpha}}S_{\alpha\dot{\alpha}}=\chi_\alpha+D_\alpha X$, with $\chi_\alpha$ and $X$ as above,
contains both stress tensor and R-symmetry current and sources 16/16 supergravity.
\subsection{Unitary SCFT irreps}
See \arxiv{1612.00809}, in Lorentzian signature;
$[j_1,\dots]_\Delta^{(R_1,\dots)}$ denotes a conformal family whose primary has Dynkin labels $\vec{j}$ and $\vec{R}$ (in $\ZZ_{\geq 0}$) for Lorentz and R-symmetry, and dimension~$\Delta$;
superconformal multiplets are long $L[\vec{j}]_\Delta^{(\vec{R})}$ with $\Delta\geq\Delta_A(\vec{j},\vec{R})$ or short $X_\ell[\vec{j}]_\Delta^{(\vec{R})}$ with $X\in\{A,B,C,D\}$ and $\Delta=\Delta_X(\vec{j},\vec{R})$ and $\ell=\ell_X(\vec{j},\vec{R})\in\ZZ_{\geq 0}$ and $\Delta_A>\Delta_B>\Delta_C>\Delta_D$.