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\begin{document}

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\section*{spin group}

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

\hypertarget{higher_spin_geometry}{}\paragraph*{{Higher spin geometry}}\label{higher_spin_geometry}

[[!include higher spin geometry - contents]]

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

[[!include group theory - contents]]

\hypertarget{higher_lie_theory}{}\paragraph*{{Higher Lie theory}}\label{higher_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{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_whitehead_tower_of_orthogonal_group}{Relation to Whitehead tower of orthogonal group}\dotfill \pageref*{relation_to_whitehead_tower_of_orthogonal_group} \linebreak
\noindent\hyperlink{ExceptionalIsomorphisms}{Exceptional isomorphisms}\dotfill \pageref*{ExceptionalIsomorphisms} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{spin_geometry}{Spin geometry}\dotfill \pageref*{spin_geometry} \linebreak
\noindent\hyperlink{in_physics}{In physics}\dotfill \pageref*{in_physics} \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}

The \textbf{spin group} $Spin(n)$ is the [[universal covering space]] of the [[special orthogonal group]] $SO(n)$. By the usual arguments it inherits a group structure for which the operations are smooth and so is a [[Lie group]] like $SO(n)$.

For special cases in low dimensions see at: [[Spin(2)]], [[Spin(3)]], [[Spin(4)]], [[Spin(5)]], [[Spin(6)]], [[Spin(7)]], [[Spin(8)]]

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

\begin{defn}
\label{QuadraticVectorSpace}\hypertarget{QuadraticVectorSpace}{}
A \emph{quadratic vector space} $(V, \langle -,-\rangle)$ is a [[vector space]] $V$ over finite [[dimension]] over a [[field]] $k$ of [[characteristic]] 0, and equipped with a symmetric [[bilinear form]] $\langle -,-\rangle \colon V \otimes V \to k$.

\end{defn}
Conventions as in (\hyperlink{Varadarajan04}{Varadarajan 04, section 5.3}).

We write $q\colon v \mapsto \langle v ,v \rangle$ for the corresponding [[quadratic form]].

\begin{defn}
\label{}\hypertarget{}{}
The \emph{[[Clifford algebra]]} $CL(V,q)$ of a quadratic vector space, def. \ref{QuadraticVectorSpace}, is the [[associative algebra]] over $k$ which is the [[quotient]]

\begin{displaymath}
Cl(V,q)
    \coloneqq
  T(V)/I(V,q)
\end{displaymath}
of the [[tensor algebra]] of $V$ by the ideal generated by the elements $v \otimes v - q(v)$.

\end{defn}
Since the [[tensor algebra]] $T(V)$ is naturally $\mathbb{Z}$-graded, the Clifford algebra $Cl(V,q)$ is naturally $\mathbb{Z}/2\mathbb{Z}$-graded.

Let $(\mathbb{R}^n, q = {\vert -\vert})$ be the $n$-dimensional [[Cartesian space]] with its canonical [[scalar product]]. Write $Cl^\mathbb{C}(\mathbb{R}^n)$ for the [[complexification]] of its [[Clifford algebra]].

\begin{prop}
\label{}\hypertarget{}{}
There exists a unique [[complex number|complex]] [[representation]]

\begin{displaymath}
Cl^{\mathbb{C}}(\mathbb{R}^n) \longrightarrow End(\Delta_n)
\end{displaymath}
of the algebra $Cl^\mathbb{C}(\mathbb{R}^n)$ of smallest [[dimension]]

\begin{displaymath}
dim_{\mathbb{C}}(\Delta_n) = 2^{[n/2]}
  \,.
\end{displaymath}
\end{prop}
\begin{defn}
\label{SpinGroup}\hypertarget{SpinGroup}{}
The [[Pin group]] $Pin(V;q)$ of a quadratic vector space, def. \ref{QuadraticVectorSpace}, is the [[subgroup]] of the [[group of units]] in the [[Clifford algebra]] $Cl(V,q)$

\begin{displaymath}
Pin(V,q) \hookrightarrow GL_1(Cl(V,q))
\end{displaymath}
on those elements which are multiples $v_1 \cdots v_{n}$ of elements $v_i \in V$ with $q(v_i) = 1$.

The [[Spin group]] $Spin(V,q)$ is the further [[subgroup]] of $Pin(V;q)$ on those elements which are even number multiples $v_1 \cdots v_{2k}$ of elements $v_i \in V$ with $q(v_i) = 1$.

Specifically, ``the'' Spin group is

\begin{displaymath}
Spin(n) \coloneqq Spin(\mathbb{R}^n)
  \,.
\end{displaymath}
\end{defn}
A \emph{[[spin representation]]} is a [[linear representation]] of the spin group, def. \ref{SpinGroup}.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

By definition the spin group sits in a [[short exact sequence]] of groups

\begin{displaymath}
\mathbb{Z}_2 \to Spin \to SO
  \,.
\end{displaymath}
\hypertarget{relation_to_whitehead_tower_of_orthogonal_group}{}\subsubsection*{{Relation to Whitehead tower of orthogonal group}}\label{relation_to_whitehead_tower_of_orthogonal_group}

The spin group is one element in the [[Whitehead tower]] of $O(n)$, which starts out like

\begin{displaymath}
\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n)
  \,.
\end{displaymath}
The [[homotopy group]]s of $O(n)$ are for $k \in \mathbb{N}$ and for sufficiently large $n$

\begin{displaymath}
\itexarray{
     \pi_{8k+0}(O) & = \mathbb{Z}_2
     \\
     \pi_{8k+1}(O) & = \mathbb{Z}_2
     \\
     \pi_{8k+2}(O) & = 0
     \\
     \pi_{8k+3}(O) & = \mathbb{Z}
     \\
     \pi_{8k+4}(O) & = 0
     \\
     \pi_{8k+5}(O) & = 0
     \\
     \pi_{8k+6}(O) & = 0
     \\
     \pi_{8k+7}(O) & = \mathbb{Z}
  }
  \,.
\end{displaymath}
By [[Whitehead tower|co-killing]] these groups step by step one gets

\begin{displaymath}
\itexarray{
     cokill\, this &&&& to\,get
     \\
     \\
     \pi_{0}(O) & = \mathbb{Z}_2 &&& SO
     \\
     \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin
     \\
     \pi_{2}(O) & = 0
     \\
     \pi_{3}(O) & = \mathbb{Z} &&& String
     \\
     \pi_{4}(O) & = 0
     \\
     \pi_{5}(O) & = 0
     \\
     \pi_{6}(O) & = 0
     \\
     \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane
  }
  \,.
\end{displaymath}
Via the [[J-homomorphism]] this is related to the [[stable homotopy groups of spheres]]:

[[!include image of J -- table]]

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

In low [[dimensions]] the [[spin groups]] happens to be [[isomorphism|isomorphic]] to various other [[classical Lie groups]]. One speaks of \emph{[[exceptional isomorphisms]]} or \emph{[[sporadic isomorphisms]]}.

See for instance (\hyperlink{Garrett13}{Garrett 13}). See also \emph{[[division algebra and supersymmetry]]}.

In the following $Sp(n)$ denotes the [[quaternionic unitary group]] in [[quaternion|quaternionic]] [[dimension]] $n$.

We have

\begin{itemize}%
\item in [[Euclidean geometry|Euclidean]] signature

\begin{itemize}%
\item $Spin(1) \simeq O(1)$


\item [[Spin(2)]] $\simeq U(1) \simeq SO(2) \simeq S^1$ ([[SO(2)]], the [[circle group]], see also at \emph{[[real Hopf fibration]]})

the projection $Spin(2)\to SO(2)$ corresponds to $S^1\stackrel{\cdot 2}{\longrightarrow} S^1$, see also at \emph{[[Theta characteristic]]}


\item [[Spin(3)]] $\simeq Sp(1) \simeq SU(2) \simeq S^3$ (the [[special unitary group]] [[SU(2)]]

the inclusion $Spin(2) \hookrightarrow Spin(3)$ corresponds to the canonical $S^1 \hookrightarrow S^3$ (see e.g. \hyperlink{GorbunovRay92}{Gorbunov-Ray 92})


\item [[Spin(4)]] $\simeq Sp(1)\times Sp(1) \simeq S^3 \times S^3$

this is given by identifying $\mathbb{R}^4 \simeq \mathbb{H}$ with the [[quaternions]] and $SU(2) \simeq S^3$ with the group of unit quternions. Then left and right quaternion multiplication gives a [[homomorphism]]

\begin{displaymath}
SU(2) \times SU(2) \longrightarrow SO(4)
\end{displaymath}
\begin{displaymath}
(g,h) \mapsto ( x \mapsto \; g^{-1}  x h )
\end{displaymath}
which is a [[double cover]] and hence exhibits the isomorphism.

In particular therefore the inclusion $Spin(3) \hookrightarrow Spin(4)$ corresponds to the [[diagonal]] $S^3 \hookrightarrow S^3 \times S^3$.

At the level of [[Lie algebras]] $\mathfrak{so}(4) \simeq \wedge^2 \mathbb{R}^4$ and the $\pm 1$-[[eigenspaces]] of the [[Hodge star operator]] $\star \colon \Wedge^2 \mathbb{R}^4 \to \mathbb{R}^4$ gives the [[direct sum]] decomposition $\mathfrak{so}(4) \simeq \mathfrak{su}(2) \oplus \mathfrak{su}(2) \simeq \mathfrak{so}(3) \oplus \mathfrak{so}(3)$


\item [[Spin(5)]] $\simeq Sp(2)$ (an indirect consequence of [[triality]], see e.g. \hyperlink{CadekVanzura97}{Čadek-Vanžura 97})


\item [[Spin(6)]] $\simeq SU(4)$ (the [[special unitary group]] \href{special+unitary+group#SU4}{SU(4)})



\end{itemize}

\item in [[Lorentz group|Lorentzian]] signature

\begin{itemize}%
\item $Spin(1,1) \simeq GL(1,\mathbb{R})$


\item $Spin(2,1) \simeq SL(2, \mathbb{R})$ -- 2d [[special linear group]] of [[real numbers]]


\item $Spin(3,1) \simeq SL(2,\mathbb{C})$ -- 2d [[special linear group]] of [[complex numbers]]


\item $Spin(4,1) \simeq Sp(1,1)$


\item $Spin(5,1) \simeq SL(2,\mathbb{H})$ -- 2d [[special linear group]] of [[quaternions]]


\item $Spin(9,1) \simeq_{in\;some\;sense} SL(2, \mathbb{O})$ -- 2d [[special linear group]] of [[octonions]]



\end{itemize}

\item in [[anti de Sitter group|anti de Sitter]] signature

\begin{itemize}%
\item $Spin(2,2) \simeq SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$


\item $Spin(3,2) \simeq Sp(4,\mathbb{R})$


\item $Spin(4,2) \simeq SU(2,2)$



\end{itemize}

\item in mixed signature

\begin{itemize}%
\item $Spin(3,3) \simeq SL(4,\mathbb{R})$ (\hyperlink{Garrett13}{Garrett 13 (2.12)})

\end{itemize}


\end{itemize}
Beyond these dimensions there are still some interesting identifications, but the situation becomes much more involved.

[[!include exceptional spinors and division algebras -- table]]

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

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

See [[spin geometry]]

\hypertarget{in_physics}{}\subsubsection*{{In physics}}\label{in_physics}

The name arises due to the requirement that the structure group of the [[tangent bundle]] of [[spacetime]] lifts to $Spin(n)$ so as to `define particles with spin'\ldots{} (Someone more awake and focused please put this into proper words!)

See [[spin structure]].

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

\begin{itemize}%
\item [[spinor]], [[spin representation]], [[spinor bundle]]

\end{itemize}
The [[Whitehead tower]] of the [[orthogonal group]] looks like

$\cdots \to$ [[fivebrane group]] $\to$ [[string group]] $\to$ \textbf{spin group} $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]].

Another extension of $SO$ is the [[spin{\tt \symbol{94}}c group]].

\begin{itemize}%
\item [[semi-spin group]]

\end{itemize}
[[!include table of orthogonal groups and related]]

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

A standard textbook reference is

\begin{itemize}%
\item [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], chapter I, section 2 of \emph{[[Spin geometry]]}, Princeton University Press (1989)

\end{itemize}
See also

\begin{itemize}%
\item [[Veeravalli Varadarajan]], section 7 of \emph{[[Supersymmetry for mathematicians]]: An introduction}, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)


\item wikipedia \emph{\href{http://en.wikipedia.org/wiki/Spin_group}{Spin group}}



\end{itemize}
Examples of sporadic (exceptional) spin group isomorphisms incarnated as [[isogenies]] onto [[orthogonal groups]] are discussed in

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


\item Vassily Gorbunov, Nigel Ray, \emph{Orientations of $Spin$ Bundles and Symplectic Cobordism}, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 ([[GorbunovRaySpinBundles.pdf:file]])



\end{itemize}
The exceptional isomorphism $Spin(5) \simeq Sp(2)$ is discussed via [[triality]] in

\begin{itemize}%
\item [[Martin Čadek]], [[Jiří Vanžura]], \emph{On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles}, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (\href{https://www.jstor.org/stable/43737249}{jstor:43737249})

\end{itemize}
Discussion of the [[cohomology]] of the [[classifying space]] $B Spin$ includes

\begin{itemize}%
\item E. Thomas, \emph{On the cohomology groups of the classifying space for the stable spinor groups}, Bol. Soc. Mat. Mexicana (2) 7 (1962) 57-69.


\item [[Harsh Pittie]], \emph{The integral homology and cohomology rings of SO(n) and Spin(n)}, Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105--153 ()



\end{itemize}
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