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\begin{document}

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\section*{Poincaré Lie algebra}

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

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{invariant_polynomials_and_chernsimons_elements}{Invariant polynomials and Chern-Simons elements}\dotfill \pageref*{invariant_polynomials_and_chernsimons_elements} \linebreak
\noindent\hyperlink{lie_algebra_valued_forms}{Lie algebra valued forms}\dotfill \pageref*{lie_algebra_valued_forms} \linebreak
\noindent\hyperlink{related_structures}{Related structures}\dotfill \pageref*{related_structures} \linebreak


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

The \emph{Poincar\'e{} Lie algebra} $\mathfrak{iso}(\mathbb{R}^{d-1,1})$ is the [[Lie algebra]] of the [[isometry group]] of [[Minkowski spacetime]]: the [[Poincaré group]]. This happens to be the [[semidirect product]] of the [[special orthogonal Lie algebra]] $\mathfrak{so}(d-1,1)$ with the the abelian translation Lie algebra $\mathbb{R}^{d-1,1}$.

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

\begin{defn}
\label{}\hypertarget{}{}
For $d \in \mathbb{N}$, write $\mathbb{R}^{d-1,1}$ for [[Minkowski spacetime]], regarded as the [[inner product space]] whose underlying [[vector space]] is $\mathbb{R}^d$ and equipped with the [[bilinear form]] given in the canonical [[linear basis]] of $\mathbb{R}^d$ by

\begin{displaymath}
\eta \coloneqq diag(-1,+1,+1, \cdots, +1)
  \,.
\end{displaymath}
The [[Poincaré group]] $Iso(\mathbb{R}^{d-1,1})$ is the [[isometry group]] of this inner product space. The \emph{Poincar\'e{} Lie algebra} $\mathfrak{iso}(\mathbb{R}^{d-1,1})$ is the [[Lie algebra]] of this [[Lie group]] (its [[Lie differentiation]])

\begin{displaymath}
\mathfrak{iso}(\mathbb{R}^{d-1,1})
  \coloneqq
  Lie(Iso(\mathbb{R}^{d-1,1}))
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The [[Poincaré group]] is the [[semidirect product group]]

\begin{displaymath}
Iso(\mathbb{R}^{d-1,1})
    \simeq
  \mathbb{R}^{d-1,1} \rtimes O(d-1,1)
\end{displaymath}
of the [[Lorentz group]] $O(d-1,1)$ (the group of [[linear map|linear]] [[isometries]] of [[Minkowski spacetime]]) with the $\mathbb{R}^d$ regarded as the [[translation group]] along itself, via the defining [[action]].

Accordingly, the Poincar\'e{} Lie algebra is the [[semidirect product Lie algebra]]

\begin{displaymath}
\mathfrak{iso}(\mathbb{R}^{d-1,1})
  \simeq
  \mathbb{R}^{d-1,1}
    \rtimes
  \mathfrak{so}^+(d-1,1)
\end{displaymath}
of the abelian Lie algebra on $\mathbb{R}^d$ with the (orthochronous) [[special orthogonal Lie algebra]] $\mathfrak{so}(d-1,1)$.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
For $\{P_a\}$ the canonical [[linear basis]] of $\mathbb{R}^d$, and for $\{L_{a b} = - L_{b a}\}$ the corresponding canonical basis of $\mathfrak{so}(d-1,1)$, then the [[Lie bracket]] in $\mathfrak{iso}(\mathbb{R}^{d-1,1})$ is given as follows:

\begin{displaymath}
\begin{aligned}
    [P_a, P_b] & = 0
    \\
    [L_{a b}, L_{c d}] 
     & = 
      \eta_{d a} L_{b c}
      -\eta_{b c} L_{a d}
      +\eta_{a c} L_{b d}
      -\eta_{d b} L_{a c}
     \\
    [L_{a b}, P_c] & = \eta_{a c} P_b -\eta_{bc} P_a
  \end{aligned}
\end{displaymath}
\end{prop}
\begin{proof}
Since [[Lie differentiation]] sees only the [[connected component]] of a [[Lie group]], and does not distinguish betwee a Lie group and any of its discrete [[covering spaces]], we may equivalently consider the Lie algebra of the [[spin group]] $Spin(d-1,1) \to SO^+(d-1,1)$ (the double cover of the [[proper orthochronous Lorentz group]]) and its [[action]] on $\mathbb{R}^{d-1,1}$.

By the discussion at \emph{[[spin group]]}, the Lie algebra of $Spin(d-1,1)$ is the Lie algebra spanned by the [[Clifford algebra]] [[bivectors]]

\begin{displaymath}
L_{a b} \leftrightarrow \Gamma_a \Gamma_b
\end{displaymath}
and its [[action]] on itself as well as on the vectors, identified with single Clifford generators

\begin{displaymath}
P_a \leftrightarrow \Gamma_a
\end{displaymath}
is given by forming [[commutators]] in the [[Clifford algebra]]:

\begin{displaymath}
[L_{a b}, P_c]
    \leftrightarrow
  \tfrac{1}{2}[\Gamma_{a b}, \Gamma_c  ]
\end{displaymath}
\begin{displaymath}
[L_{a b}, L_{c d}]
  \leftrightarrow
  \tfrac{1}{2}[\Gamma_{a b}, \Gamma_{c d}  ]
  \,.
\end{displaymath}
Via the Clifford relation

\begin{displaymath}
\Gamma_a \Gamma_b + \Gamma_b \Gamma_a = -2 \eta_{a b}
\end{displaymath}
this yields the claim.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Dually, the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{iso}(\mathbb{R}^{d-1})$ is generated from $\mathbb{R}^{d,1}$ and $\wedge^2 \mathbb{R}^{d,1}$. For $\{t_a\}$ the standard basis of $\mathbb{R}^{d-1,1}$ we write $\{\omega^{a b}\}$ and $\{e^a\}$ for these generators. With $(\eta_{a b})$ the components of the [[Minkowski metric]] we write

\begin{displaymath}
\omega^{a}{}_b \coloneqq \omega^{a c}\eta_{c b}
  \,.
\end{displaymath}
In terms of this the CE-differential that defines the Lie algebra structure is

\begin{displaymath}
d_{CE} \colon \omega^{a b} = \omega^a{}_c \wedge \omega^{c b}
\end{displaymath}
\begin{displaymath}
d_{CE} \colon e^a \mapsto \omega^{a}{}_b  \wedge t^b
\end{displaymath}
\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology}

We discuss some elements in the [[Lie algebra cohomology]] of $\mathfrak{iso}(d-1,1)$.

The canonical degree-3 $\mathfrak{so}(d-1,1)$-cocycle is

\begin{displaymath}
\omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a
  \in 
  CE(\mathfrak{iso}(d-1,1))
  \,.
\end{displaymath}
The \emph{volume cocycle} is the [[volume form]]

\begin{displaymath}
vol = \epsilon_{a_1 \cdots a_{d}} e^{a_1} \wedge \cdots \wedge e^{a_d} \in CE(\mathfrak{iso}(d-1,1))
  \,.
\end{displaymath}
\hypertarget{invariant_polynomials_and_chernsimons_elements}{}\subsubsection*{{Invariant polynomials and Chern-Simons elements}}\label{invariant_polynomials_and_chernsimons_elements}

With the basis elements $(e^a, \omega^{a b})$ as above, denote the shifted generators of the [[Weil algebra]] $W(\mathfrak{iso}(d-1,1))$ by $\theta^a$ and $r^{a b}$, respectively.

We have the [[Bianchi identity]]

\begin{displaymath}
d_W : r^{a b} 
   \mapsto
   \omega^{a c} \wedge R_c{}^d
   - 
   R^{a c} \wedge \omega_c{}^b
\end{displaymath}
and

\begin{displaymath}
d_W : \theta^a 
   \mapsto
    \omega^a{}_b \theta^b
    - 
    R^{a}{}_b e^b
   \,.
\end{displaymath}
The element $\eta_{a b} \theta^a \wedge \theta^b \in W(\mathfrak{iso}(d-1,1))$ is an [[invariant polynomial]]. A [[Chern-Simons element]] for it is $cs = \eta_{a b} e^a \wedge \theta^b$. So this transgresses to the trivial cocycle.

Another invariant polynomial is $r^{a b} \wedge r_{a b}$. This is the [[Killing form]] of $\mathfrak{so}(d-1,1)$. Accordingly, it transgresses to a multiple of $\omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a$.

This is the first in an infinite series of Pontryagin invariant polynomials

\begin{displaymath}
P_n 
  :=
  r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3}
  \wedge \cdots \wedge r^{a_n}{}_{a_1}
  \,.
\end{displaymath}
There is also an infinite series of mixed invariant polynomials

\begin{displaymath}
C_{2n + 2}
  :=
  \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3}
  \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge \theta^{a_n}
  \,.
\end{displaymath}
[[Chern-Simons element]]s for these are

\begin{displaymath}
B_{2n + 1}
  :=
  \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3}
  \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge e^{a_n}
  \,.
\end{displaymath}
\hypertarget{lie_algebra_valued_forms}{}\subsubsection*{{Lie algebra valued forms}}\label{lie_algebra_valued_forms}

A [[Lie algebra-valued form]] with values in $\mathfrak{iso}(d-1,1)$

\begin{displaymath}
\Omega^\bullet(X) \leftarrow W(\mathfrak{iso}(d-1,1)) : (E,\Omega)
\end{displaymath}
is

\begin{itemize}%
\item a [[vielbein]] $E$ on $X$;


\item a ``[[spin connection]]'' $\Omega$ on $X$.



\end{itemize}
The [[curvature]] 2-form $(T, R)$ consists of

\begin{itemize}%
\item the [[torsion of a metric connection|torsion]] $T = d E + [\Omega \wedge E]$;


\item the [[Riemannian curvature]] $R = d \Omega + [\Omega \wedge \Omega]$.



\end{itemize}
If the torsion vanishes, then $\Omega$ is a [[Levi-Civita connection]] for the [[metric]] $E^a \otimes E^b \eta_{a b}$ defined by $E$.

The [[volume form]] is the image of the \hyperlink{VolumeCocycle}{volume cocycle}

\begin{displaymath}
\Omega^\bullet(X)
  \stackrel{(E,\Omega)}{\leftarrow}
  W(\mathfrak{iso}(d-1,1))
  \stackrel{vol}{\leftarrow}
  W(b^{d-1} \mathbb{R})
  :
  vol(E)
  \,.
\end{displaymath}
We have

\begin{displaymath}
vol(E) = \epsilon_{a_1 \cdots a_d} E^{a_1} \wedge \cdots \wedge E^{a_d}
  \,.
\end{displaymath}
If the torsion vanishes, this is indeed a closed form.

\hypertarget{related_structures}{}\subsection*{{Related structures}}\label{related_structures}

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


\item [[super Poincaré Lie algebra]]



\end{itemize}
[[!redirects Poincaré Lie algebras]]

[[!redirects Poincare Lie algebra]] [[!redirects Poincaré Lie algebra]]



\end{document}