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Closed FLRW metric, Lie group of nonzero quaternions
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%%%   date      = "12-May-2006",
%%%   time      = "12:30:44 EST",
%%%   author    = "Vladimir Trifonov",
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%%%                Cumberland, RI 02864,
%%%                USA",
%%%   email     = "trifonov@member.ams.org",
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\documentclass[12pt, twoside]{article} 
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\markboth{\ \hrulefill V. Trifonov}{ Natural geometry of nonzero quaternions \hrulefill\ } 
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\begin{document}
\title{Natural geometry of nonzero quaternions} 
\author{Vladimir Trifonov \\ AMS \\ 
trifonov@member.ams.org} 
\date{}\maketitle 
\begin{abstract} 
It is shown that the group of nonzero quaternions carries a family of natural closed  Friedmann-Lema\^{\i}tre-Robertson-Walker metrics. \end{abstract} 

\section*{Introduction.} The quaternion algebra $\HH$ is one of the most important and well-studies objects in mathematics (e.~g. \cite{Wid02} and references therein) and physics (e.~g. \cite{Adl95} and references therein). It has a natural Hermitian form which induces a Euclidean scalar product on its additive vector space $S_{\HH}$. There is also a family of natural indefinite scalar products of signature 2 on $S_{\HH}$ \cite{Tri95}, induced by the structure tensor $\bo{H}$ of the quaternion algebra. This result came out of a study of relationship between natural metric properties of \emph{unital} algebras and internal logic of topoi they generate. It was shown in \cite{Tri95} that if the logic of a topos is bivalent Boolean then the generating algebra is isomorphic to the quaternion algebra with a family of natural scalar product of signature 2. Such scalar products can be defined on any linear algebra over a field $\F$. In this note we show that for a unital algebra these scalar products can be naturally extended over the Lie group of its invertible elements, producing a family of \emph{principal} \emph{metrics}. In particular, for the quaternion algebra, these metrics are closed Friedmann-Lema\^{\i}tre-Robertson-Walker. 
\begin{rem} Some of the notations and definitions are slightly nonstandard. We use the $[\begin{smallmatrix} m\\n \end{smallmatrix}]$ device to denote tensor ranks; for example a one-form is a $[\begin{smallmatrix} 0\\1 \end{smallmatrix}]$-tensor. For clarity of the exposition we use $\Box$ at the end of a \emph{Proof}, and each \emph{Remark} ends the sign appearing at the end of this line. \hfill $\Diamond$ \end{rem}.
\begin{defn} An $\F$-\emph{algebra}, $A$, is an ordered pair $(S_A, \bo{A})$, where $S_A$ is a vector space over a field $\F$, and $\bo{A}$ is a $[\begin{smallmatrix} 1\\2 \end{smallmatrix}]$-tensor  on $S_A$, called the \emph{structure} \emph{tensor} of $A$. Each vector $\bo{a}$ of $S_A$ is called an \emph{element} of $A$, denoted $\bo{a} \in A$.  The \emph{dimensionality} of $A$ is that of $S_A$. \end{defn} 
\begin{rem} This is an unconventional definition of a linear algebra over $\F$. Indeed, the tensor $\bo{A}$ induces a binary operation $S_A \times S_A \to S_A$, called the \emph{multiplication} of $A$: to each pair of vectors $(\bo{a}, \bo{b})$ the tensor $\bo{A}$ associates a vector $\bo{ab} : S^*_A \to \F$, such that $(\bo{ab})(\tilde{\bo{\tau}}) =  \bo{A}(\tilde{\bo{\tau}}, \bo{a}, \bo{b}), \forall \tilde{\bo{\tau}} \in S^*_A$. An $\F$-algebra with an associative multiplication is called \emph{associative}. An element $\bo{\imath}$, such that $\bo{a\imath} = \bo{\imath a} =\bo{a}, \forall \bo{a} \in A$ is called an \emph{identity} of $A$.  \hfill $\Diamond$ \end{rem} 
\begin{defn} For an $\F$-algebra $A$ and each nonzero one-form $\tilde{\bo{\tau}} \in S^*_A$, the tensor $\bo{A}[\tilde{\bo{\tau}}] := \tilde{\bo{\tau}} \lrcorner \bo{A}$ is called a \emph{principal} \emph{scalar} \emph{product} on $A$, just in case it is symmetric, where $\lrcorner$ denotes contraction on the first index. \end{defn} 
\begin{defn} For each $\F$-algebra $A = (S_A, \bo{A})$, an $\F$-algebra $[A] = (S_A,$ $[\bo{A}])$, with the structure tensor defined by  \begin{displaymath} [\bo{A}](\tilde{\bo{\tau}}, \bo{a}, \bo{b}) := \bo{A}(\tilde{\bo{\tau}}, \bo{a}, \bo{b}) - \bo{A}(\tilde{\bo{\tau}}, \bo{b}, \bo{a}), \forall \tilde{\bo{\tau}} \in S^*_A, \bo{a}, \bo{b} \in S_A , \end{displaymath} 
is called the \emph{commutator} algebra of $A$. \end{defn}
\begin{defn} A finite dimensional associative $\R$-algebra with an identity is called a \emph{unital} algebra. \end{defn}
\begin{lem} The set $\mc{A}$ of all invertible elements of a unital algebra $A$ is a Lie group with respect to the multiplication of $A$ whose Lie algebra of $\mc{A}$ is the commutator algebra $[A]$. \end{lem} 
\begin{proof} See, for example, \cite{Pos82} for a proof of this simple lemma. \end{proof} 
\begin{rem} For each basis $(\bo{e}_j)$ on the vector space $S_A$ of a unital algebra, there is a natural basis field on $\mc{A}$, namely the basis $(\hat{\bo{e}}_j)$ of left invariant vector fields generated by $(\bo{e}_j)$, associating to each point $\bo{a} \in A$ a basis $(\hat{\bo{e}}_j)(\bo{a})$ on the tangent space $T_{\bo{a}}\mc{A}$. We call $(\hat{\bo{e}}_j)$ a \emph{proper} \emph{frame} \emph{generated} \emph{by} $(\bo{e}_j)$. The value of $(\hat{\bo{e}}_j)$ at $\bo{a}$ is referred to as a \emph{proper} \emph{basis} (at $\bo{a}$) generated by $(\bo{e}_j)$, and denoted $(\hat{\bo{e}}_j)(\bo{a})$. In particular, $(\hat{\bo{e}}_j)(\bo{\imath})$, the proper basis at the identity generated by $(\bo{e}_j)$ coincides with $(\bo{e}_j)$. \hfill $\Diamond$ \end{rem}
\begin{defn} For a unital algebra $A$, let $(\hat{\bo{e}}_j)$ be a  proper frame on $\mc{A}$, generated by a basis $(\bo{e}_j)$ on $S_A$. The \emph{structure} \emph{field} of the Lie group $\mc{A}$ is a tensor field $\bo{\mc{A}}$ on $\mc{A}$, assigning to each point $\bo{a} \in \mc{A}$ a $[\begin{smallmatrix} 1\\2 \end{smallmatrix}]$-tensor $\bo{\mc{A}}(\bo{a})$ on $T_a\mc{A}$, with components $\mc{A}^i_{jk}(\bo{a})$ in the basis  $(\hat{\bo{e}}_j)(\bo{a})$, defined by 
\begin{displaymath} \mc{A}^i_{jk}(\bo{a}) := A^i_{jk} , \quad \forall \bo{a} \in \mc{A} ,  \end{displaymath} where $A^i_{jk}$ are the components of the structure tensor $\bo{A}$ in the basis $(\bo{e}_j)$. \end{defn} 
\begin{defn} For a unital algebra $A$ and each $\bo{a} \in \mc{A}$, an $\F$-algebra $\mc{A}(\bo{a}) = (T_{\bo{a}}\mc{A}, \bo{\mc{A}(a)})$ is called the \emph{tangent} \emph{algebra} of the Lie group $\mc{A}$ at $\bo{a}$. \end{defn}
\begin{rem} It is easy to see that for each $\bo{a} \in \mc{A}$, the tangent algebra $\mc{A}(\bo{a})$ is isomorphic to $A$; in particular, each $A(\bo{a})$ is unital. \hfill $\Diamond$ \end{rem} 
\begin{defn} For a unital algebra $A$ and a twice differentiable real function $\mc{T}$ on the Lie group $\mc{A}$, a \emph{principal} \emph{metric} \emph{on} $\mc{A}$ is a $[\begin{smallmatrix} 0\\2 \end{smallmatrix}]$-tensor field $\bo{\mc{T}} := d\mc{T} \lrcorner \bo{\mc{A}}$, where $d\mc{T}$ is the gradient of $\mc{T}$, such that $\bo{\mc{T}}(\bo{a})$ is a principal scalar product on $\mc{A}(\bo{a}), \forall \bo{a} \in \mc{A}$. \end{defn} 
\section{Quaternion algebra.}
\begin{defn} A four dimensional $\R$-algebra, $\HH = (S_{\HH}, \bo{H})$, is called a \emph{quaternion} \emph{algebra} (with \emph{quaternions} as its elements), if there is a basis on $S_{\HH}$, in which the components of the structure tensor $\bo{H}$ are given by the entries of the following matrices, 
\begin{multline} \label{QST} H^0_{\alpha \beta} = 
\begin{pmatrix} 1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1 \end{pmatrix},\ 
H^1_{\alpha \beta} = \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-
1&0 \end{pmatrix}, \\ H^2_{\alpha \beta} = \begin{pmatrix} 
0&0&1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0 \end{pmatrix},\ H^3_{\alpha 
\beta} = \begin{pmatrix} 0&0&0&1\\0&0&1&0\\0&-1&0&0\\1&0&0&0 
\end{pmatrix}. \end{multline} We refer to such a basis as \emph{canonical}. \end{defn} 
\begin{rem} The vectors of the canonical basis are denoted $\bo{1}$, $\bo{i}$, $\bo{j}$, $\bo{k}$. A quaternion algebra is unital, with the first vector  of the canonical basis, $\bo{1}$, as its identity. Since $(\bo{1}$, $\bo{i}$, $\bo{j}$, $\bo{k})$ is a basis on a real vector space, any quaternion $\bo{a}$ can be presented as $a^0\bo{1} + a^1\bo{i} + a^2\bo{j} + a^3\bo{k}, a^{\beta} \in \R$. A quaternion $\bar{\bo{a}} = a^0\bo{1} - a^1\bo{i} - a^2\bo{j} - a^3\bo{k}$ is called \emph{conjugate} to $\bo{a}$. We refer to $a^0$ and $a^p\bo{i}_p$ as the \emph{real} and  \emph{imaginary} \emph{part} of $\bo{a}$, respectively. Quaternions of the form $a^0\bo{1}$ are in one-to-one correspondence with real numbers, which is often denoted, with certain notational abuse, as $\R \subset \HH$. \hfill $\Diamond$ \end{rem}
\begin{rem} \label{SO(3)} A linear transformation $S_{\HH} \to S_{\HH}$ with the following components in the canonical basis, 
\begin{displaymath} \begin{pmatrix} 1 & 0 \\ 0& \bo{\mf{B}} \end{pmatrix}, \bo{\mf{B}} \in SO(3), \end{displaymath} 
takes $(\bo{1}$, $\bo{i}$, $\bo{j}$, $\bo{k})$ to a basis $(\bo{i}_{\beta})$ in which the components \eqref{QST} of the structure tensor will \emph{not} change, and neither will the multiplicative behavior of vectors of $(\bo{i}_{\beta})$. Thus, we have a class of canonical bases parameterized by elements of $SO(3)$. \hfill $\Diamond$ \end{rem}
\begin{thm} Every principal scalar product on $\HH$ is of signature $2$. \end{thm} 
\begin{proof} For the quaternion algebra the components of the structure tensor $\bo{H}$ in a canonical basis are given by \eqref{QST}. 
A one-form $\tilde{\bo{\tau}}$ on $S_{\HH}$ with components $\tilde{\tau}_{\beta}$ in (the dual of) a canonical basis $(\bo{i}_{\beta})$ contracts with the structure tensor into a $[\begin{smallmatrix} 0\\2 \end{smallmatrix}]$-tensor on $S_{\HH}$ with the following components in the basis $(\bo{i}_{\beta})$: 
\begin{displaymath} \begin{pmatrix} 
\tilde{\tau}_0& \tilde{\tau}_1& \tilde{\tau}_2& \tilde{\tau}_3\\ 
\tilde{\tau}_1&-\tilde{\tau}_0& \tilde{\tau}_3&-\tilde{\tau}_2\\ 
\tilde{\tau}_2&-\tilde{\tau}_3&-\tilde{\tau}_0& \tilde{\tau}_1\\ 
\tilde{\tau}_3& \tilde{\tau}_2&-\tilde{\tau}_1&-\tilde{\tau}_0 \end{pmatrix}. \end{displaymath} 
The only way to make this symmetric is to put $\tilde{\tau}_1=-\tilde{\tau}_1$, $\tilde{\tau}_2=-\tilde{\tau}_2$,  $\tilde{\tau}_3=-\tilde{\tau}_3$, which yields $\tilde{\tau}_1=\tilde{\tau}_2=\tilde{\tau}_3=0$: 
\begin{equation} \label{PSP} A[\tilde{\bo{\tau}}]_{\alpha \beta} = \begin{pmatrix} 
\tilde{\tau}_0& 0& 0& 0\\ 
0&-\tilde{\tau}_0& 0&0\\ 
0&0&-\tilde{\tau}_0& 0\\ 
0& 0&0&-\tilde{\tau}_0 \end{pmatrix}. \end{equation} . \end{proof}
\section{Natural structures on $\mc{H}$.} There is a class of canonical bases   
on $S_{\HH}$ (see Remark \ref{SO(3)}) whose members differ from one another by a rotation in the hyperplane of pure 
imaginary quaternions. Each canonical basis $(\bo{i}_{\beta})$ induces a \emph{canonical}  
coordinate system $(w$, $x$, $y$, $z)$ on $S_{\HH}$, considered as a (linear) manifold, and 
therefore also on its submanifold $\mc{H}$ of nonzero quaternions: a quaternion $\bo{a} = a^{\beta}\bo{i}_{\beta}$ 
is assigned coordinates $(w = a^0$, $x = a^1$, $y = a^2$, $z = a^3)$. This coordinate 
system covers both $S_{\HH}$ and $\mc{H}$ with a single patch. Since $\bo{0} 
\notin \mc{H}$, at least one of the coordinates is always nonzero for any point $\bo{a} \in 
\mc{H}$. For a differentiable function $R : \R \to \R \setminus \{0\}$ there is a system of natural spherical coordinates  $(\eta$, $\chi$, $\theta$, $\phi)$ on $\mc{H}$, related to the canonical coordinates by 
\begin{multline*} 
w = R(\eta)\cos(\chi), \quad x = R(\eta)\sin(\chi)\sin(\theta)\cos(\phi), \\
y = R(\eta)\sin(\chi)\sin(\theta)\sin(\phi), \quad z = R(\eta)\sin(\chi)\cos(\theta) . 
\end{multline*} 
Each canonical basis $(\bo{i}_{\beta})$ 
can be considered a basis on the vector space of the Lie algebra of $\mc{H}$, i.~e., 
the tangent space $T_{\bo{1}}\mc{H} \cong S_{\HH}$ to $\mc{H}$ at the point $(1$, $0$, 
$0$, $0)$, the identity of the group $\mc{H}$. There are several natural basis fields on 
$\mc{H}$ induced by each basis $(\bo{i}_{\beta})$. First of all, there are two coordinate 
basis fields, the \emph{canonical} \emph{frame}, $(\partial_w$, $\partial_x$, $\partial_y$, $\partial_z)$ and the corresponding \emph{spherical} \emph{frame}  $(\partial_{\eta}$, $\partial_{\chi}$, $\partial_{\theta}$, $\partial_{\phi})$. We also have a noncoordinate basis field, the \emph{proper} \emph{frame}  $(\hat{\bo{\imath}}_{\beta})$, of left invariant vector fields on $\mc{H}$, induced by the canonical basis. For each frame $(\bo{f}_{\beta})$ on $\mc{H}$, its value at $\bo{a}$, i.~e., a basis on $T_{\bo{a}}\mc{H}$, is denoted $(\bo{f}_{\beta})(\bo{a})$. A left invariant vector field $\hat{\bo{u}}$ on $\mc{H}$, generated by a vector $\bo{u} \in S_{\HH}$ with components $(u^{\beta})$ in a canonical basis, associates to each point $\bo{a} \in \mc{H}$ with coordinates $(w$, $x$, $y$, $z)$ a vector $\hat{\bo{u}}(\bo{a}) \in T_{\bo{a}}\mc{H}$ with the components $\hat{u}^{\beta}(\bo{a}) = (\bo{a}\bo{u})^{\beta}$ in the basis $(\partial_w$, $\partial_x$, $\partial_y$, $\partial_z)(\bo{a})$ on $T_{\bo{a}}\mc{H}$: 
\begin{multline} \label{LVFIELDS} 
\hat{u}^0(\bo{a}) = wu^0 - xu^1 - yu^2 - zu^3 , \quad \hat{u}^1(\bo{a}) = wu^1 + xu^0 + yu^3 - zu^2 , \\ 
\hat{u}^2(\bo{a}) = wu^2 - xu^3 + yu^0 + zu^1 , \quad \hat{u}^3(\bo{a}) = wu^3 + xu^2 - yu^1 + zu^0 . 
\end{multline} 
The system \eqref{LVFIELDS} contains sufficient information to compute transformation between the frames. 
For example, the transformation between the spherical and proper frames is given by 
\begin{displaymath} \begin{pmatrix} 
R/\dot{R} & 0 & 0 & 0\\ 
0 & \sin{\theta}\cos{\phi} & \sin{\theta}\sin{\phi} & \cos{\theta} \\ 
0 & \frac{\cos{\chi}\cos{\theta}\cos{\phi}+\sin{\chi}\sin{\phi}}{\sin{\chi}} & 
\frac{\cos{\chi}\cos{\theta}\sin{\phi}+\sin{\chi}\cos{\phi}}{\sin{\chi}} & 
\frac{\cos{\chi}\sin{\theta}}{\sin{\chi}}\\ 
0 & \frac{\sin{\chi}\cos{\theta}\cos{\phi}-\cos{\chi}\sin{\phi}}{\sin{\chi}\sin{\theta}} & 
\frac{\sin{\chi}\cos{\theta}\sin{\phi}+\cos{\chi}\cos{\phi}}{\sin{\chi}\sin{\theta}} & -1 
\end{pmatrix}, \end{displaymath} where $\dot{R} := \frac{dR}{d{\eta}} : \R \to \R \setminus \{0\}$. 
\begin{defn} A Lorentzian metric on a four dimensional manifold is called \emph{closed} \emph{FLRW} (Friedmann-Lema\^{\i}tre-Robertson-Walker) if there is a coordinate system $(\eta, \chi, \theta, \phi)$, such that in the corresponding coordinate frame the components of the metric are given by the entries of the following matrix: \begin{displaymath} 
\genfrac{}{}{0pt}{3}{+}{-} \begin{pmatrix} 
1&0&0&0\\ 0&-\mf{a}^2&0&0\\ 0&0&-\mf{a}^2{\sin^2(\chi)} &0\\ 
0&0&0&-\mf{a}^2{\sin^2(\chi)} {\sin^2(\theta)} \end{pmatrix}, \end{displaymath} where $\mf{a} : \R \to \R$, referred to as the \emph{scale} \emph{factor}, is a function of $\eta$ only. \end{defn}
\section{Principal metrics on $\mc{H}$.}
\begin{thm} Every principal metric of $\HH$ is closed FLRW. \end{thm} 
\begin{proof} Let $\tilde{\bo{\tau}}$ and $(\bo{i}_{\beta})$ be a one-form and a canonical basis on $S_{\HH}$, respectively. For each point $\bo{a} \in \mc{H}$ the $\R$-algebra $\mc{H}(\bo{a}) := (T_{\bo{a}}\mc{H}, \bo{\mc{H}(a)})$ is the tangent algebra, at $\bo{a}$, of the Lie group $\mc{H}$. For each $\bo{a} \in \mc{H}$ the components of the structure tensor  $\bo{\mc{H}(a)}$ and a principal scalar product, $\bo{\mc{H}}[\tilde{\bo{\tau}}]$ of $\mc{H}(\bo{a})$ in the basis  $(\bo{\hat{\imath}}_{\beta})(\bo{a})$ are given by \eqref{QST} and  \eqref{PSP}, respectively. Therefore, the components of a principal metric, $\bo{\mc{T}}$, in the proper frame $(\bo{\hat{\imath}}_{\beta})$ must have the form 
\begin{equation} \begin{pmatrix} 
\tilde{\tau}& 0& 0& 0\\ 
0&-\tilde{\tau}& 0&0\\ 
0&0&-\tilde{\tau}& 0\\ 
0& 0&0&-\tilde{\tau} \end{pmatrix}, \end{equation} 
for some function $\tilde{\tau} : \mc{H} \to \R \setminus \{0\}$. In other words, any principal metric on $\mc{H}$ is obtained by contraction of a one-form field $\tilde{\bo{\tau}}$, whose components in $(\bo{\hat{\imath}}_{\beta})$ are $(\tilde{\tau}, 0, 0, 0)$, with the structure field $\bo{\mc{H}}$. This one-form is exact, i.~e., there exists a twice differentiable function $\mc{T}$, such that $d\mc{T} = \tilde{\bo{\tau}}$. In the spherical frame with $R(\eta) = \exp(\eta)$ the components of $\tilde{\bo{\tau}}$ are also $(\tilde{\tau}$, $0$, $0$, $0)$ , and,   
\begin{equation} \label{TIME} d\mc{T}_0 = \frac{\partial \mc{T}}{\partial \eta} = \tilde{\tau}, \quad d\mc{T}_1 = \frac{\partial \mc{T}}{\partial \chi} = d\mc{T}_2 = \frac{\partial \mc{T}}{\partial \theta} = d\mc{T}_3 = \frac{\partial \mc{T}}{\partial \phi} = 0 . \end{equation} 
It follows from \eqref{TIME} that both $\mc{T}$ and $\tilde{\tau}$ depend on $\eta$ only. Since $\frac{\partial \mc{T}}{\partial \eta}$ is differentiable, $\tilde{\tau}$ must be at least continuous. Therefore $\tilde{\tau}$ cannot change sign, because $\tilde{\tau}(\eta) \neq 0, \forall \eta \in \R$. Computing the components of the principal metric $\bo{\mc{T}}$ in the spherical frame we get
\begin{displaymath} \mc{T}_{\alpha \beta} = \begin{pmatrix} 
\tau(\eta)(\frac{\dot{R}}{R})^2&0&0&0\\0&-\tau(\eta)&0&0\\ 
0&0&-\tau(\eta){\sin^2(\chi)} &0\\ 
0&0&0&-\tau(\eta){\sin^2(\chi)} {\sin^2(\theta)} \end{pmatrix}.  \end{displaymath}  
If $\tau(\eta) > 0$, we take $R(\eta)$ such that $\tau(\eta)(\frac{\dot{R}}{R})^2 = 
1$, which yields \begin{equation} \label{+R} R(\eta) = 
exp{\int\frac{d\eta}{\genfrac{}{}{0pt}{3}{+}{-}\sqrt{\tau(\eta)}}} \quad . \end{equation} 
In other words, with $R(\eta)$ satisfying \eqref{+R}, the components of the principal metric in the spherical frame are 
\begin{displaymath} 
\mc{T}_{\alpha \beta} = \begin{pmatrix} 
1&0&0&0\\ 0&-\mf{a}^2&0&0\\ 0&0&-\mf{a}^2{\sin^2(\chi)} &0\\ 
0&0&0&-\mf{a}^2{\sin^2(\chi)} {\sin^2(\theta)} \end{pmatrix}, \end{displaymath} 
where the scale factor $\mf{a}(\eta) := {\sqrt{\tau(\eta)}}$. \par 
If $\tau(\eta) < 0$, similar considerations show that the metric is also closed FLRW with the scale factor $\mf{a}(\eta) := {\sqrt{-\tau(\eta)}}$. \end{proof} 
\begin{cor} $\mc{T}$ is a monotonous function of $\eta$ for each principal metric $\bo{\mc{T}}$ of $\mc{H}$. \end{cor} 
Thus the natural geometry of the group of nonzero quaternions $\mc{H}$ is defined by a family of closed  Friedmann-Lema\^{\i}tre-Robertson-Walker metrics with monotonous scale factors. 
\begin{thebibliography}{99} 
\bibitem[Adl95]{Adl95} S. L. Adler, \emph{Quaternionic Quantum Mechanics and Quantum 
Fields}, Oxford University Press, Oxford, UK (1995). 
\bibitem[Wid02]{Wid02} D. Widdows, \emph{Quaternionic Algebraic Geometry}, PhD Thesis, St Annes's College, Oxford, UK (2002). 
\bibitem[Pos82]{Pos82} {\font \t = wncyr10 scaled 1200 \t M. M. Postnikov, Gruppy i algebry Li, 
Nauka, Moskva (1982)}.
\bibitem[Tri95]{Tri95} V. Trifonov, \emph{Europhysics Letters}, 32 (8), 1995, pp. 621-626. 
\end{thebibliography}
\end{document}
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