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\begin{document}

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\section*{Spin(4)}

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

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{spin_geometry}{}\paragraph*{{Spin geometry}}\label{spin_geometry}

[[!include higher spin geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ExceptionalIsomorphisms}{Exceptional isomorphisms}\dotfill \pageref*{ExceptionalIsomorphisms} \linebreak
\noindent\hyperlink{euler_class_and_pontryagin_class}{Euler class and Pontryagin class}\dotfill \pageref*{euler_class_and_pontryagin_class} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

$Spin(4)$ is the [[spin group]] in dimension 4, the [[double cover]] of [[SO(4)]].

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

\hypertarget{ExceptionalIsomorphisms}{}\subsubsection*{{Exceptional isomorphisms}}\label{ExceptionalIsomorphisms}

Let $\mathbb{H}$ be the [[real vector space]] underlying the [[quaternions]]. Notice that [[Spin(3)]] is the group of unit quaternions under quaternion multiplication

\begin{displaymath}
Spin(3) \simeq S(\mathbb{H})
  \,.
\end{displaymath}
This induces a [[group homomorphism]]

\begin{equation}
\itexarray{
    Spin(3) \times Spin(3)
    &\longrightarrow&
    O(4)
    \\
    (e_1, e_2)
    &\mapsto&
    \big(
      q 
      \mapsto
       e_1 \cdot q \cdot \overline{e_2}
    \big)
  }
\label{Spin3SquareToO4}\end{equation}
\begin{prop}
\label{ExceptionalIsoWithSpin3TimesSpin3}\hypertarget{ExceptionalIsoWithSpin3TimesSpin3}{}
The [[group homomorphism]] \eqref{Spin3SquareToO4} is a [[double cover]] and hence exhibits an [[isomorphism]] between [[Spin(4)]] and the [[direct product group]] of [[Spin(3)]] with itself:

\begin{equation}
\vartheta \;\colon\;
  Spin(3) \times Spin(3)
  \overset{\simeq}{\longrightarrow}
   Spin(4)
\label{Spin3TimesSpin3IsSpin4}\end{equation}
Since the [[action]] of [[Spin(3)]] on the [[imaginary part|imaginary]] [[quaternions]] $\mathbb{H}_{im} \simeq_{\mathbb{R}} \mathbb{R}^3$ is the [[conjugation action]] by unit quaternions, it follows in particular, that the canonical inclusion of [[Spin(3)]] into [[Spin(4)]] is given by the [[diagonal]] morphsm with respect to the identification \eqref{Spin3SquareIsSpin4}:

\begin{equation}
\itexarray{
    Spin(3)
    &\hookrightarrow&
    Spin(4)
    \\
    &
    {}_{\mathllap{\Delta}}\searrow
    &
    \Big\downarrow^{ \mathrlap{\simeq} }
    \\
    && Spin(3) \times Spin(3)

  }
\label{Spin3Diagonally}\end{equation}
\end{prop}
(e.g. \hyperlink{Garrett13}{Garrett 13})

In summary:

\begin{prop}
\label{}\hypertarget{}{}
There is a [[commuting diagram]] of [[Lie groups]] of the form

\begin{displaymath}
\itexarray{
    ( q_1, q_2 )
    &\mapsto&
    (x \mapsto q_1 \cdot x \cdot \overline{q}_2) 
    \\
    Sp(1) \times Sp(1) 
    &\overset{\simeq}{\longrightarrow}&
    Spin(4)
    \\
    \big\downarrow
    &&
    \big\downarrow
    \\
    Sp(1)\cdot Sp(1)
    &\overset{\simeq}{\longrightarrow}&
    SO(4)
  }
\end{displaymath}
where

\begin{enumerate}%
\item in the top left we have [[Sp(1)]] = [[Spin(3)]],


\item in the top right we have [[Spin(4)]],


\item in the bottom left we have [[Sp(n).Sp(1)|Sp(1).Sp(1)]]


\item in the bottom right we have [[SO(4)]]


\item the horizontal morphism assigns the [[conjugation action]] of unit [[quaternions]], as indicated,


\item the right vertical morphism is the defining [[double cover]],


\item the left vertical morphism is the defining [[quotient group]]-projection.



\end{enumerate}
\end{prop}
\begin{remark}
\label{ExceptionalIsoViaDynkinDiagrams}\hypertarget{ExceptionalIsoViaDynkinDiagrams}{}
\textbf{(exceptional isomorphism via Dynkin diagrams)}

Under the [[classification of simple Lie groups]] via [[Dynkin diagrams]], and via the further exceptional isomorphism [[Spin(3)]] $\simeq$ [[SU(2)]], the exceptional isomorphism \eqref{Spin3TimesSpin3IsSpin4} corresponds to the coincidence of the [[D3]] with the [[A3]] diagrams, both with their central node removed:

$\backslash$begin\{tikzpicture\}

$\backslash$node at (0,1.4) \{$\mathrm{Spin}(4)$\}; $\backslash$node at (3.4,1.4) \{$\mathrm{SU}(2) \times \mathrm{SU}(2)$\};

$\backslash$node at (1.7,1.4) \{$\simeq$\};

$\backslash$node (center) at (0,0) \{\}; $\backslash$node (topright) at (30:1) \{\}; $\backslash$node (left) at (180-30:1) \{\}; $\backslash$node (botright) at (0,-1) \{\}; \%$\backslash$node (D5) at (-2,0) \{\}; \%$\backslash$node (D6) at (-3,0) \{\};

$\backslash$draw\href{center}{draw=lightgray, fill=lightgray} circle (.1); $\backslash$draw\href{topright}{fill=black} circle (.1); $\backslash$draw\href{botright}{draw=lightgray, fill=lightgray} circle (.1); $\backslash$draw\href{left}{fill=black} circle (.1);

\%$\backslash$draw\href{D5}{fill=black} circle (.1); \%$\backslash$draw\href{D6}{fill=black} circle (.1);

$\backslash$draw\href{center}{lightgray} to (topright); $\backslash$draw\href{center}{lightgray} to (botright); $\backslash$draw\href{center}{lightgray} to (left);

\%$\backslash$draw (D5) to (left); \%$\backslash$draw (D6) to (D5);

$\backslash$begin\{scope\}shift=\{(3.4,0)\} $\backslash$node (center) at (0,0) \{\}; $\backslash$node (left) at (-1,0) \{\}; $\backslash$node (right) at (+1,0) \{\};

$\backslash$draw\href{center}{draw=lightgray, fill=lightgray} circle (.1); $\backslash$draw\href{left}{fill=black} circle (.1); $\backslash$draw\href{right}{fill=black} circle (.1);

$\backslash$draw\href{center}{lightgray} to (left); $\backslash$draw\href{center}{lightgray} to (right); $\backslash$end\{scope\} $\backslash$end\{tikzpicture\}

\end{remark}


\hypertarget{euler_class_and_pontryagin_class}{}\subsubsection*{{Euler class and Pontryagin class}}\label{euler_class_and_pontryagin_class}

\begin{prop}
\label{IntegralCohomologyOfClassifyingSpace}\hypertarget{IntegralCohomologyOfClassifyingSpace}{}
\textbf{([[integral cohomology]] of [[classifying space]]/[[universal characteristic classes]])}

The [[integral cohomology|integral]] [[cohomology ring]] of the [[classifying space]] of [[Spin(3)]] is [[polynomial ring|freely generated]] from $1/4$th of the [[first Pontryagin class]]:

\begin{displaymath}
H^\bullet
  \big(
    B Spin(3),
    \mathbb{Z}
  \big)
  \;\simeq\;
  \mathbb{Z}
  \big[
    \tfrac{1}{4}p_1
  \big]
\end{displaymath}
Moreover, the [[integral cohomology|integral]] [[cohomology ring]] of the [[classifying space]] of [[Spin(4)]] is [[polynomial ring|freely generated]] from the [[first fractional Pontryagin class]] $\tfrac{1}{2}p_1$ and the combination $\tfrac{1}{2}\big( \chi + \tfrac{1}{2}p_1 \big)$, where $\chi$ is the [[Euler class]]:

\begin{displaymath}
H^\bullet
  \big(
    B Spin(4),
    \mathbb{Z}
  \big)
  \;\simeq\;
  \mathbb{Z}
  \big[
    \tfrac{1}{2}p_1
    \,,
    \tfrac{1}{2}\big(  \chi+ \tfrac{1}{2}p_1 \big)
  \big]
\end{displaymath}
Finally, under the [[exceptional isomorphism]] \eqref{Spin3SquareToO4} $\vartheta \;\colon\; Spin(3) \times Spin(3) \overset{\simeq}{\to} Spin(4)$ these classes are related by

\begin{displaymath}
\begin{aligned}
    \vartheta^\ast
    \left(
      \tfrac{1}{2}p_1
    \right)
    & 
    =
    \phantom{-}
    \tfrac{1}{4}p_1 \otimes 1
    +
    1 \otimes \tfrac{1}{4} p_1
    \\
    \vartheta^\ast
    \Big(
      \tfrac{1}{2}
      \big(
        \chi 
        + 
        \tfrac{1}{2} p_1
      \big)
    \Big)
    & 
    = 
    \phantom{-}
    \phantom{
      1 \otimes \tfrac{1}{4}p_1
      +
    }
    1 \otimes \tfrac{1}{4}p_1
    \\
    \text{hence}
    \phantom{AAAA}
    \vartheta^\ast\big( \chi \big)
    \phantom{
          + 
          \tfrac{1}{2} p_1
        \big)
      \Big)
    }
    & =
    -
    \tfrac{1}{4}p_1 \otimes 1
    +
    1 \otimes \tfrac{1}{4} p_1
    
  \end{aligned}
\end{displaymath}
Therefore, under the canonical [[diagonal]] inclusion $\iota \colon Spin(3) \overset{\Delta}{\hookrightarrow} Spin(3) \times Spin(3) \simeq Spin(4)$ \eqref{Spin3Diagonally} we have

\begin{displaymath}
\begin{aligned}
    \iota^\ast
    \left(
      \tfrac{1}{2}p_1
    \right)
    & 
    =
    \tfrac{1}{2}p_1
    \\
    \iota^\ast
    \big(
      \chi
    \big)
    & 
    =
    0
  \end{aligned}
\end{displaymath}
\end{prop}
(e.g. \hyperlink{CadekVanzura98}{Čadek-Vanžura 98, Lemma 2.1})

linebreak

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

[[!include low dimensional rotation groups -- table]]

$\backslash$linebreak

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

\begin{itemize}%
\item [[Martin Čadek]], [[Jiří Vanžura]], \emph{On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes}, Colloquium Mathematicum 1998, 76 (2), pp 213-228 (\href{http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-cmv76z2p213bwm}{web})


\item [[Paul Garrett]], \emph{Sporadic isogenies to orthogonal groups}, July 2013 (\href{http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf}{pdf})



\end{itemize}
[[!redirects Spin4]]



\end{document}