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\begin{document}

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\section*{Sp(n).Sp(1)}

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

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

[[!include group 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{as_the_effective_quotient_of__acting_on_}{As the effective quotient of $Sp(n)\times Sp(1)$ acting on $\mathbb{H}^n$}\dotfill \pageref*{as_the_effective_quotient_of__acting_on_} \linebreak
\noindent\hyperlink{lift_to_}{Lift to $Sp(n) \times Sp(1)$}\dotfill \pageref*{lift_to_} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{SO4}{$Sp(1)\cdot Sp(1)$ is $SO(4)$}\dotfill \pageref*{SO4} \linebreak
\noindent\hyperlink{Spinc}{$Spin(n)\cdot Spin(2)$ is $Spin^c(n)$}\dotfill \pageref*{Spinc} \linebreak
\noindent\hyperlink{Triality}{Triality}\dotfill \pageref*{Triality} \linebreak
\noindent\hyperlink{Sp1Sp1Sp1}{$Sp(1)Sp(1)Sp(1) = Spin(4)\cdot Spin(3)$}\dotfill \pageref*{Sp1Sp1Sp1} \linebreak
\noindent\hyperlink{SpinGrassmannians}{Spin-Grassmannians}\dotfill \pageref*{SpinGrassmannians} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{_2}{$Sp(n)\cdot Sp(1)$}\dotfill \pageref*{_2} \linebreak
\noindent\hyperlink{ReferencesSp2Sp1}{$Sp(2)\cdot Sp(1)$}\dotfill \pageref*{ReferencesSp2Sp1} \linebreak
\noindent\hyperlink{_4}{$Spin(n_1)\cdot Spin(n_2)$}\dotfill \pageref*{_4} \linebreak
\noindent\hyperlink{ReferencesSpin4Spin3}{$Sp(1)Sp(1)Sp(1) \simeq Spin(4)\cdot Spin(3)$}\dotfill \pageref*{ReferencesSpin4Spin3} \linebreak


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

The [[Lie group]] denoted $Sp(n).Sp(1)$ (\hyperlink{Alekseevskii68}{Alekseevskii 68}, \hyperlink{Gray69}{Gray 69}) or just $Sp(n)Sp(1)$ is the [[quotient group]] of the [[direct product group]] of the given [[quaternion unitary groups]] by their [[diagonal]] [[center]] [[cyclic group of order 2]].

A [[smooth manifold]] of [[dimension]] $4n$ with [[G-structure]] for this group $G = Sp(n).Sp(1)$ is a [[quaternion-Kähler manifold]].

Similarly, for $Spin(n_1)$, $Spin(n_2)$ [[spin groups]] in some dimension, the group denoted $Spin(n_1) \cdot Spin(n_2)$ or just $Spin(n_1)Spin(n_2)$ is the [[quotient group]] of the [[direct product group]] $Spin(n_1) \times Spin(n_2)$ by the [[diagonal]] [[center]] [[cyclic group of order 2]].

These products $G_1 \cdot G_2$ are examples of [[central products of groups]].

$\backslash$linebreak

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

\begin{defn}
\label{SpnSp1}\hypertarget{SpnSp1}{}
For $n \in \mathbb{N}$ with $n \geq 2$, the [[Lie group]] denoted $Sp(n).Sp(1)$ or just $Sp(n)Sp(1)$ is the [[quotient group]] of the [[direct product group]] $Sp(n) \times Sp(1)$ of [[quaternion unitary groups]] $Sp(n)$ (in particular $Sp(1) \simeq$ [[Spin(3)]]) by the [[diagonal]] [[center]] [[cyclic group of order 2]] $\mathbb{Z}_2$:

\begin{displaymath}
Sp(n).Sp(1)
  \;\coloneqq\;
  \big(
    Sp(n)
     \times
    Sp(1)
  \big)/_{diag}\mathbb{Z}_2
\end{displaymath}
hence the [[quotient group]] by the [[subgroup]]

\begin{equation}
\mathbb{Z}_2
  \;\simeq\;
  \big\{ 
    (1,1), (-1,-1)
  \big\}
  \hookrightarrow
  Sp(n) \times Sp(1)
  \,.
\label{DiagonalCenter}\end{equation}
\end{defn}
(e.g. \hyperlink{CadekVanzura97}{Čadek-Vanžura 97, Sec. 2})

A similar definition yields

\begin{defn}
\label{Spinn1Spinn2}\hypertarget{Spinn1Spinn2}{}
Write

\begin{displaymath}
Spin(n_1) \cdot Spin(n_2)
  \;\coloneqq\;
  \big(
    Spin(n_1)
    \times 
    Spin(n_2)
  \big)/\mathbb{Z}_2
\end{displaymath}
for the [[quotient group]] of the [[direct product group]] of [[spin groups]] by their [[diagonal]] [[subgroup]]

\begin{displaymath}
\mathbb{Z}_2
  \;\simeq\;
  \big\{
    (1,1), (-1,-1)
  \big\}
  \;\hookrightarrow\;
  Spin(n_1) \times Spin(n_1)
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{McInnes99a}{McInnes 99a, p. 9}, \hyperlink{HilgertNeeb12}{Hilgert-Neeb 12, Prop. 17.3.1})

Sometimes one sees the notation further generalized to include cases such as

\begin{itemize}%
\item $Spin(n) \cdot U(1) \simeq Spin(n)\cdot Spin(2) \simeq$ [[Spin{\tt \symbol{94}}c]],

\end{itemize}
see Example \ref{SpinnSpin2IsSpinc} below.

$\backslash$linebreak

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

\hypertarget{as_the_effective_quotient_of__acting_on_}{}\subsubsection*{{As the effective quotient of $Sp(n)\times Sp(1)$ acting on $\mathbb{H}^n$}}\label{as_the_effective_quotient_of__acting_on_}

The [[direct product group]] $Sp(n) \times Sp(1)$ has a canonical [[action]] on the [[quaternion]] [[vector space]] $\mathbb{H}^n$, where the factor [[Sp(n)]] acts as $2 \times 2$ [[quaternion unitary group|quaternion unitary]] [[matrix multiplication]] from the left, and $Sp(1)$ acts by [[diagonal]] $1 \times 1$ matrix action on each $\mathbb{H}$-summand from the right.

For instance for $n = 2$ this action controls the [[quaternionic Hopf fibration]] and its $Sp(2)$ [[equivariant function|equivariance]] (see \href{quaternionic+Hopf+fibration#EquivariantStructure}{there}).

But this action is not an [[effective group action]]: Precisely the diagonal center \eqref{DiagonalCenter} acts trivially.

There is then a [[commuting diagram]] of [[Lie groups]]

\begin{equation}
\itexarray{
    Sp(2) \times Sp(1)
    &\longrightarrow&
    Spin(8)
    \\
    \big\downarrow && \big\downarrow
    \\
    Sp(2) \cdot Sp(1)    
    &\longrightarrow&
    SO(8)
  }
\label{CompatibilityDiagram}\end{equation}
with the horizontal maps being [[group homomorphisms]] to [[Spin(8)]] and [[SO(8)]], respectively, the left morphism being the defining [[quotient]] projection and the right morphism the [[double cover]] morphism that defines the [[spin group]].

(e.g. \hyperlink{CadekVanzura97}{Čadek-Vanžura 97, p. 4})

\hypertarget{lift_to_}{}\subsubsection*{{Lift to $Sp(n) \times Sp(1)$}}\label{lift_to_}

(\hyperlink{MarchiafavaRomani76}{Marchiafava-Romani 76}, \hyperlink{Salamon82}{Salamon 82, around Def. 2.1})

(\ldots{})

$\backslash$linebreak

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

\hypertarget{SO4}{}\subsubsection*{{$Sp(1)\cdot Sp(1)$ is $SO(4)$}}\label{SO4}

The case of $Sp(n)\cdot Sp(1)$ for $n = 1$ is special, as in this case the canonical inclusion $Sp(n)\cdot Sp(1) \hookrightarrow SO(4n)$ becomes an [[isomorphism]]

\begin{displaymath}
Sp(1)\cdot Sp(1)
  \;\simeq\;
  SO(4)
\end{displaymath}
with the [[special orthogonal group]] [[SO(4)]], and hence the compatibility diagram \eqref{CompatibilityDiagram} now exhibits at the top the [[exceptional isomorphism]] $Sp(1) \times Sp(1) \simeq$ [[Spin(4)]] (see \href{Spin4#ExceptionalIsomorphisms}{there})

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}
$\backslash$linebreak

\hypertarget{Spinc}{}\subsubsection*{{$Spin(n)\cdot Spin(2)$ is $Spin^c(n)$}}\label{Spinc}

\begin{example}
\label{SpinnSpin2IsSpinc}\hypertarget{SpinnSpin2IsSpinc}{}
For $n \in \mathbb{N}$, group $Sp(n) \cdot Sp(2)$ in Def. \ref{Spinn1Spinn2} is the group otherwise known as [[spin{\tt \symbol{94}}c|spin{\tt \symbol{94}}c(n)]]:

\begin{displaymath}
Spin(n)\cdot Spin(2)
  \;\simeq\;
  Spin^c(n)
  \,.
\end{displaymath}
This is due to the identification of the [[double cover]] by [[Spin(2)]] of [[SO(2)]] with the [[real Hopf fibration]] (\href{Spin2#Spin2OverSO2IsRealHopfFibration}{this Prop}), which identifies $Spin(2) \simeq U(1)$ compatible with the subgroupinclusion of $\mathbb{Z}_2$.

\end{example}
(See also e.g. \hyperlink{Gompf97}{Gompf 97, p. 2})

\hypertarget{Triality}{}\subsubsection*{{Triality}}\label{Triality}

\begin{prop}
\label{Spin3DotSpin5SubgroupsInSO8}\hypertarget{Spin3DotSpin5SubgroupsInSO8}{}
\textbf{([[Spin(5).Spin(3)]]-[[subgroups]] in [[SO(8)]])}

The [[direct product group]] [[SO(3)]] $\times$ [[SO(5)]] together with the groups [[Sp(2).Sp(1)]] and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into [[SO(8)]], form 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[subgroups]] inside [[SO(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

$\backslash$begin\{center\} $\backslash$begin\{xymatrix\} Sp(2)$\backslash$cdot Sp(1) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}dr{\tt \symbol{94}}\{$\backslash$iota'\} $\backslash$ \& SO(8) \& SO(3) $\backslash$times SO(5) $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}l{\tt \symbol{94}}-\{$\backslash$iota\} $\backslash$ Sp(1) $\backslash$cdot Sp(2) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}ur\emph{-\{ $\backslash$iota'` \} $\backslash$end\{xymatrix\} $\backslash$end\{center\}}

\end{prop}
(\hyperlink{Kollross02}{Kollross 02, Prop. 3.3 (3)})

Similarly:

\begin{prop}
\label{Spin3DotSpin5SubgroupsInSpin8}\hypertarget{Spin3DotSpin5SubgroupsInSpin8}{}
\textbf{([[Spin(5).Spin(3)]]-[[subgroups]] in [[Spin(8)]])}

The groups [[Spin(5).Spin(3)]], [[Sp(2).Sp(1)]] and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into [[Spin(8)]], form 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[subgroups]] inside [[Spin(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

$\backslash$begin\{center\} $\backslash$begin\{xymatrix\} Sp(2)$\backslash$cdot Sp(1) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}dr{\tt \symbol{94}}\{$\backslash$iota'\} $\backslash$ \& Spin(8) \& Spin(3) $\backslash$cdot Spin(5) $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}l{\tt \symbol{94}}-\{$\backslash$iota\} $\backslash$ Sp(1) $\backslash$cdot Sp(2) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}ur\emph{-\{ $\backslash$iota'` \} $\backslash$end\{xymatrix\} $\backslash$end\{center\}}

\end{prop}
(\hyperlink{CadekVanzura97}{Čadek-Vanžura 97, Sec. 2})

In summary:

$\backslash$begin\{xymatrix\} $\backslash$mathrm\{Sp\}(1) \{$\backslash$cdot\} $\backslash$mathrm\{Sp\}(2) $\backslash$ar@\{=\}dddddr $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}ddrr|-\{ $\backslash$iota' \} $\backslash$ $\backslash$ \&\& $\backslash$mathrm\{Spin\}(8) $\backslash$ar@\{-\guillemotright{}\}dddddr \&\& $\backslash$mathrm\{Spin\}(3) \{$\backslash$cdot\} $\backslash$mathrm\{Spin\}(5) $\backslash$ar@\{-\guillemotright{}\}dddddr $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}ll|-\{$\backslash$iota\} $\backslash$ $\backslash$ $\backslash$mathrm\{Sp\}(2) \{$\backslash$cdot\} $\backslash$mathrm\{Sp\}(1) $\backslash$ar@\{=\}dddddr $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}uurr |\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\guillemotleft{}\{ $\backslash$phantom\{AA $\backslash$atop AA\} \} |\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}\guillemotright{}{\tt \symbol{62}}\{ $\backslash$iota'` \} $\backslash$ \& $\backslash$mathrm\{Sp\}(1) \{$\backslash$cdot\} $\backslash$mathrm\{Sp\}(2) $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}ddrr|-\{ $\backslash$iota' \} $\backslash$ $\backslash$ \&\&\& $\backslash$mathrm\{SO\}(8) \&\& $\backslash$mathrm\{SO\}(3) $\backslash$times $\backslash$mathrm\{SO\}(5) $\backslash$ar@\{\_\{(\}-{\tt \symbol{62}}\}ll|-\{$\backslash$iota\} $\backslash$ $\backslash$ \& $\backslash$mathrm\{Sp\}(2) \{$\backslash$cdot\} $\backslash$mathrm\{Sp\}(1) $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}uurr|-\{ $\backslash$iota'` \} $\backslash$end\{xymatrix\}

\hypertarget{Sp1Sp1Sp1}{}\subsubsection*{{$Sp(1)Sp(1)Sp(1) = Spin(4)\cdot Spin(3)$}}\label{Sp1Sp1Sp1}

\begin{example}
\label{Spin4Spin3}\hypertarget{Spin4Spin3}{}
\textbf{([[Spin(4).Spin(3)]])}

The group

\begin{displaymath}
Spin(4)\cdot Spin(3)
  \;\coloneqq\;
  \big(  
    Spin(4) \times Spin(3)
  \big)/\mathbb{Z}_2
\end{displaymath}
is the [[quotient group]] of the [[direct product group]] of [[Spin(4)]] with [[Spin(3)]] by the [[subgroup]]

\begin{equation}
\mathbb{Z}_2
  \;\simeq\;
  \big\{ 
    (1,1), (-1,-1)
  \big\}
  \hookrightarrow
  Spin(4) \times Spin(3)
  \,.
\label{Spin4Spin3DiagonalCenter}\end{equation}
Due to the exception [[isomorphism]] [[Spin(4)]] $\simeq$ [[Spin(3)]] $\times$ [[Spin(3)]] (\href{Spin4#ExceptionalIsoWithSpin3TimesSpin3}{this Prop.}) this is [[isomorphism|isomorphic]] to the [[quotient group]] of the [[direct product group|direct product]] of 3 copies of [[Sp(1)]] $\simeq$ [[Spin(3)]] with itself

\begin{displaymath}
Spin(4)\cdot Spin(3)
  \;\simeq\;
  Sp(1)Sp(1)Sp(1)
  \;\coloneqq\;
  \big( Spin(3) \times Spin(3) \times Spin(3)\big)/_{diag} \mathbb{Z}_2
\end{displaymath}
by the triple diagonal center

\begin{equation}
\mathbb{Z}_2
  \;\simeq\;
  \big\{ 
    (1,1,1), (-1,-1,-1)
  \big\}
  \hookrightarrow
  Spin(3) \times Spin(3) \times Spin(3)
  \,.
\label{Spin3Spin3Spin3DiagonalCenter}\end{equation}
\end{example}
See the references \hyperlink{ReferencesSpin4Spin3}{below}.

\begin{example}
\label{}\hypertarget{}{}
The [[coset space]] of [[Sp(2).Sp(1)]] (Def. \ref{SpnSp1}) by [[Sp(1)Sp(1)Sp(1)]] (Def. \ref{Spin4Spin3}) is the [[4-sphere]]:

\begin{displaymath}
\frac{
    Sp(2)\cdot Sp(1)
  }
  {
    Sp(1)Sp(1)Sp(1)
  }
  \;\simeq\;
  S^4
  \,.
\end{displaymath}
This follows essentially from the [[quaternionic Hopf fibration]] and its $Sp(2)$-[[equivariant function|equivariance]]\ldots{}

\end{example}
(e.g. \hyperlink{BettiolMendes15}{Bettiol-Mendes 15, (3.1), (3.2), (3.3)})

\hypertarget{SpinGrassmannians}{}\subsubsection*{{Spin-Grassmannians}}\label{SpinGrassmannians}

We have the following [[coset spaces]] of [[spin groups]] by dot-products of Spin groups as above:

\begin{displaymath}
Spin(7)/ \big(  Spin(4)\cdot Spin(3) \big)
  \;\simeq\;
  SO(7) / \big( SO(4) \times SO(3)  \big)
  \;\simeq\;
  Gr(4, 7)
\end{displaymath}
is the space of [[Cayley 4-planes]] ([[Cayley 4-form]]-[[calibrated submanifolds]] in 8d [[Euclidean space]]). This happens to also be [[homeomorphism|homeomorphic]] to just the plain [[Grassmannian]] of 4-planes in 7d (recalled e.g. in \hyperlink{OrneaPiccini00}{Ornea-Piccini 00, p. 1}).

Similarly,

\begin{displaymath}
Spin(6)/ \big(  Spin(3)\cdot Spin(3) \big)
  \;\simeq\;
  SU(6)/ SO(4)
\end{displaymath}
is the Grassmannian of those [[Cayley 4-planes]] that are also [[special Lagrangian submanifolds]] (\href{Cayley+form#BBMOOY96}{BBMOOY 96, p. 7 (8 of 17)}).

Moreover,

\begin{displaymath}
Spin(8)/ \big(  Spin(5)\cdot Spin(3) \big)
  \;\simeq\;
  Gr(3, 8)
\end{displaymath}
is the [[Grassmannian]] of 3-planes in 8d. (\hyperlink{CadekVanzura97}{Cadek-Vanzura 97, Lemma 2.6}).

$\backslash$linebreak

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

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

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

\hypertarget{_2}{}\subsubsection*{{$Sp(n)\cdot Sp(1)$}}\label{_2}

Very early appearances of the notation $Sp(n)\cdot Sp(1)$ are mostly in discussions of [[Berger's theorem]] for exceptional [[holonomy]]:

\begin{itemize}%
\item [[Alfred Gray]], \href{https://projecteuclid.org/euclid.mmj/1029000212}{\emph{A Note on Manifolds Whose Holonomy Group is a Subgroup of Sp(n) $\cdot$ Sp(1)}}, Michigan Math. J. Volume 16, Issue 2 (1969), 125-128.


\item [[Dmitri Alekseevskii]], \href{https://link.springer.com/article/10.1007%2FBF01075943}{\emph{Riemannian spaces with exceptional holonomy groups}}, Functional Analalysis and its Applications (1968) 2: 97.



\end{itemize}
However, the even earlier paper:

\begin{itemize}%
\item Joseph Wolff, \emph{Complex homogeneous contact manifolds and quaternionic symmetric spaces}, Journal of Mathematics and Mechanics, vol. 14 (1965), pp. 1033-1048.

\end{itemize}
describes this construction as a ``local [[direct product]]'' of [[topological groups]] and applies it to the classification of [[quaternionic manifolds]]. The notation in the classical paper of \hyperlink{}{Bonan} for this group is $V_{4n} [Sp(n) \otimes_\mathbf{H} Sp(1)]$.

Of early algebraic interest is the structure theory article:

\begin{itemize}%
\item Stefano Marchiafava, Giuliano Romani, \emph{Sul classificante del gruppo $Sp(n) \cdot Sp(1)$}, Annali di Matematica Pura ed Applicata December 1976, Volume 110, Issue 1, pp 295–319 (\href{https://doi.org/10.1007/BF02418010}{doi:10.1007/BF02418010})

\end{itemize}
More on the [[cohomology]] of $Sp(n)\cdot Sp(1)$ and its [[classifying space]]:

\begin{itemize}%
\item Stefano Marchiafava, Giuliano Romani, \emph{Alcune osservazioni sui sottogruppi abeliani del gruppo $Sp(n)\cdot Sp(1)$}, Annali di Matematica 1977 (\href{https://doi.org/10.1007/BF02413792}{doi:10.1007/BF02413792})


\item [[Paolo Piccinni]], Giuliano Romani, \emph{A generalization of symplectic Pontrjagin classes to vector bundles with structure group $Sp(n)\cdot Sp(1)$}, Annali di Matematica pura ed applicata (1983) 133: 1 (\href{https://doi.org/10.1007/BF01766008}{doi:10.1007/BF01766008})


\item [[Paolo Piccinni]], \emph{Vector fields and characteristic numbers on hyperkàhler and quaternion Kâhler manifolds}, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1992) Volume: 3, Issue: 4, page 295-298 (\href{https://eudml.org/doc/244204}{dml:244204})


\item [[Dmitri Alekseevskii]] S. Marchiafava, \emph{Quaternionic structures on a manifold and subordinated structures}, Annali di Matematica pura ed applicata (1996) 171: 205 (\href{https://doi.org/10.1007/BF01759388}{doi:10.1007/BF01759388})



\end{itemize}
Discussion of the lift to $Sp(n) \times Sp(1)$ appears in

\begin{itemize}%
\item S. Marchiafava, G. Romani, \emph{Sui fibrati con struttura quaternionale generalizzata}, Ann. Mat. Pura Appl. (IV) CVII, 131-157 (1976) (\href{https://doi.org/10.1007/BF02416470}{doi:10.1007/BF02416470})


\item [[Simon Salamon]], around Def. 2.1 in \emph{Quaternionic Kähler manifolds}, Invent Math (1982) 67: 143. (\href{https://doi.org/10.1007/BF01393378}{doi:10.1007/BF01393378})



\end{itemize}
\hypertarget{ReferencesSp2Sp1}{}\subsubsection*{{$Sp(2)\cdot Sp(1)$}}\label{ReferencesSp2Sp1}

Articles dealing specifically with the group $Sp(2)\cdot Sp(1)$:

\begin{itemize}%
\item [[Martin Čadek]], [[Jiří Vanžura]], Section 2 of \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})


\item [[Martin Čadek]], [[Jiří Vanžura]], \emph{Almost quaternionic structures on eight-manifolds}, Osaka J. Math. Volume 35, Number 1 (1998), 165-190 (\href{https://projecteuclid.org/euclid.ojm/1200787905}{euclid:1200787905})


\item Andreas Kollross, Prop. 3.3 of \emph{A Classification of Hyperpolar and Cohomogeneity One Actions}, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 (\href{https://www.jstor.org/stable/2693761}{jstor:2693761})



\end{itemize}
See also the references at \emph{[[quaternion-Kähler manifold]]}.

\hypertarget{_4}{}\subsubsection*{{$Spin(n_1)\cdot Spin(n_2)$}}\label{_4}

A textbook occurrence of dot notation for general spin groups, $Spin(n_1)\cdot Spin(n_2)$, appears in

\begin{itemize}%
\item Joachim Hilgert, [[Karl-Hermann Neeb]], Prop. 17.3.1 \emph{Structure and Geometry of Lie Groups}, Springer Monographs in Mathematics, Springer-Verlag New York, 2012 (\href{https://link.springer.com/book/10.1007/978-0-387-84794-8}{doi:10.1007/978-0-387-84794-8})

\end{itemize}
The identification of $Spin \dot Spin(2)$ with [[Spin{\tt \symbol{94}}c]] appears for instance in

\begin{itemize}%
\item [[Robert Gompf]], \emph{$Spin^c$ structures and homotopy equivalences}, Geom. Topol. 1 (1997) 41-50 (\href{https://arxiv.org/abs/math/9705218}{arXiv:math/9705218})

\end{itemize}
Discussion of central product spin groups as [[subgroups]] of [[semi-spin groups]] (motivated by analysis of the [[gauge groups]] and [[Green-Schwarz anomaly cancellation]] of [[heterotic string theory]]) is in

\begin{itemize}%
\item [[Brett McInnes]], p. 9 of \emph{The Semispin Groups in String Theory}, J. Math. Phys. 40:4699-4712, 1999 (\href{https://arxiv.org/abs/hep-th/9906059}{arXiv:hep-th/9906059})


\item [[Brett McInnes]], \emph{Gauge Spinors and String Duality}, Nucl. Phys. B577:439-460, 2000 (\href{https://arxiv.org/abs/hep-th/9910100}{arXiv:hep-th/9910100})



\end{itemize}
As such these also appear as [[U-duality groups]] and their [[subgroups]], e.g.

\begin{itemize}%
\item [[Arjan Keurentjes]], p. 10 of \emph{The topology of U-duality (sub-)groups}, Class.Quant.Grav. 21 (2004) 1695-1708 (\href{https://arxiv.org/abs/hep-th/0309106}{arXiv:hep-th/0309106})

\end{itemize}
\hypertarget{ReferencesSpin4Spin3}{}\subsubsection*{{$Sp(1)Sp(1)Sp(1) \simeq Spin(4)\cdot Spin(3)$}}\label{ReferencesSpin4Spin3}

The group $Spin(4)\cdot Spin(3) \simeq (Spin(3))^3/_{diag} \mathbb{Z}_2$ (Example \ref{Spin4Spin3}) is discussed in the following, largely in describing the Grassmannian of [[Cayley 4-planes]], see \href{Cayley+form#GrassmannianOfCayley4Planes}{there}:

\begin{itemize}%
\item Wu-Chung Hsiang, Wu-Yi Hsiang, Tables A of \emph{Differentiable Actions of Compact Connected Classical Groups: II}, Annals of Mathematics Second Series, Vol. 92, No. 2 (1970), pp. 189-223 (\href{https://www.jstor.org/stable/1970834}{jstor:1970834})


\item [[Reese Harvey]], [[H. Blaine Lawson]], theorem 1.38 of \emph{Calibrated geometries}, Acta Math. Volume 148 (1982), 47-157 (\href{https://projecteuclid.org/euclid.acta/1485890157}{Euclid:1485890157})


\item [[Robert Bryant]], [[Reese Harvey]], (3.19) in \emph{Submanifolds in Hyper-Kähler Geometry}, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 1-31 (\href{https://www.jstor.org/stable/1990911}{jstor:1990911})


\item Herman Gluck, Dana Mackenzie, Frank Morgan, (5.20) in \emph{Volume-minimizing cycles in Grassmann manifolds}, Duke Math. J. Volume 79, Number 2 (1995), 335-404 (\href{https://projecteuclid.org/euclid.dmj/1077285156}{euclid:1077285156})


\item Megan M. Kerr, Lemma 6.2 of \emph{Some New Homogeneous Einstein Metrics on Symmetric Spaces}, Transactions of the American Mathematical Society, Vol. 348, No. 1 (1996), pp. 153-171 (\href{https://www.jstor.org/stable/2155169}{jstor:2155169})


\item [[Katrin Becker]], [[Melanie Becker]], [[David Morrison]], [[Hirosi Ooguri]], Y. Oz, Z. Yin, (3.5) of \emph{Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds}, Nucl. Phys. B480:225-238, 1996 (\href{https://arxiv.org/abs/hep-th/9608116}{arXiv:hep-th/9608116})


\item [[Victor Kac]], A.V. Smilga, around (1.10) in \emph{Vacuum structure in supersymmetric Yang-Mills theories with any gauge group}, in \emph{[[The Many Faces of the Superworld]], pp. 185-234 World Scientific (2000)} (\href{https://arxiv.org/abs/hep-th/9902029}{arXiv:hep-th/9902029}, \href{https://doi.org/10.1142/9789812793850_0014}{doi:10.1142/9789812793850\_0014})


\item Liviu Ornea, [[Paolo Piccinni]], \emph{Cayley 4-frames and a quaternion-Kähler reduction related to Spin(7)}, Proceedings of the International Congress of Differential Geometry in the memory of A. Gray, held in Bilbao, Sept. 2000 (\href{https://arxiv.org/abs/math/0106116}{arXiv:math/0106116})


\item Karsten Grove, Burkhard Wilking, Wolfgang Ziller, p. 30 of \emph{Positively Curved Cohomogeneity One Manifolds and 3-Sasakian Geometry} (\href{https://arxiv.org/abs/math/0511464}{arXiv:math/0511464})


\item Renato G. Bettiol, Ricardo A. E. Mendes, \emph{Flag manifolds with strongly positive curvature}, Math. Z. 280 (2015), no. 3-4, 1031-1046 (\href{https://arxiv.org/abs/1412.0039}{arXiv:1412.0039})


\item Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Prop. 2.2 in \emph{Clifford systems in octonionic geometry} (\href{https://arxiv.org/abs/1511.06239}{arXiv:1511.06239})



\end{itemize}
Discussion of $Sp(1)\cdot Sp(1) \cdot Sp(1)$ in the context of [[super Lie algebras]] and [[superconformal geometry|superconformal symmetry]] is in:

\begin{itemize}%
\item [[Peter Freund]], p. 634 of \emph{World topology and gauged internal symmetries}, Proc. 19th Int. Conf. High Energy Physics, Tokyo 1978 (\href{http://inspirehep.net/record/137780/}{spire:137780}, \href{https://cds.cern.ch/record/870701/files/c78-08-23-p617.pdf}{pdf})

\end{itemize}
and possibly with the $\mathbb{Z}_2$-quotient not made explicit:

\begin{itemize}%
\item [[Peter Goddard]] (auth.), [[Peter Freund]], K. T. Mahanthappa, p. 128 of \emph{Superstrings}, NATO ASI Series 175, Springer 1988


\item Kazuo Hosomichi, Sangmin Lee, Sungjay Lee, Jaemo Park, slide 13 of \emph{New Superconformal Chern-Simons Theories} (\href{http://www3.ic.ac.uk/pls/portallive/docs/1/46083696.PDF}{pdf})



\end{itemize}
[[!redirects Sp(2).Sp(1)]] [[!redirects Spin(5).Spin(3)]]

[[!redirects Spn.Sp1]]

[[!redirects SpnSp1]]

[[!redirects Sp(1)Sp(1)Sp(1)]] [[!redirects Sp(1).Sp(1).Sp(1)]]

[[!redirects central product spin group]] [[!redirects central product spin groups]]

[[!redirects Spin(n).Spin(m)]]

[[!redirects Spin(n1).Spin(n2)]]

[[!redirects Spin(3)Spin(3)Spin(3)]] [[!redirects Spin(3).Spin(3).Spin(3)]]

[[!redirects Spin(4).Spin(3)]]



\end{document}