Is this a one dimensional Lorentz Boost? And can you have a 1-d Boost without group structure? Someone has claimed that he has constructed a quaternion representation of the one dimensional (along the x axis) Lorentz Boost.
His quaternion Lorentz Boost is $v'=hvh^*+ 1/2( [hhv]^*-[h^*h^*v^*]^*)$ where h is (sinh(x),cosh(x),0,0). He derived this odd transform by substituting the hyperbolic sine and cosine for the sine and cosine in the usual unit quaternion rotation $v'=hvh^*$ and then subtracting out unwanted factors. You can see the short "proof" here:  
http://visualphysics.org/preprints/qmn10091026
I have argued that, whereas the quaternion rotations form a group, his newly devised transform probably does not. Two transformations of the form $v'=hvh^*+ 1/2( [hhv]^*-[h^*h^*v^*]^*)$ almost certainly do not make another transformation of the form $v'=fvf^*+ 1/2( [ffv]^*-[f^*f^*v^*]^*)$ He has not attempted to prove that and he won't because he thinks it is unnecessary.
He has responded that, even if I'm correct, he still has created a Lorentz Boost despite the fact that he has not created a group. I've argued that that it is essential that two Lorentz Boosts make another Lorentz Boost. Without that group structure there is no Boost.
Is this correct? 
Can you have a Lorentz Boost along the x axis without having the group structure Boost+Boost=Boost? (for the case of Boosts along the x axis. I know that, for boosts in different directions, Boost+Boost=Rotation)
 A: This is really just a matter of deciding if his idea to call this a "Lorentz boost" is justified or not. It's a choice of terminology, not a fixed mathematical idea that needs a mathematician to decide whether it is right or wrong.
For regular Minkowski space, the composition of two boosts need not be a pure boost: the composition generally incorporates a spatial rotation. So I don't see any good reason that boosts should be required to form a group (although all of the Lorentz transformations form a group.)
Edit: After rereading I see that you probably know that, but are just questioning whether or not boosts composing to a nonboost would occur in a 1-dimensional setting. Unless his definition is a special case or version of higher dimensional cases, this could be debatable. Be sure to post in the physics forum too!
A: Boost along the $x$ direction.
The result is 
$X' = \Lambda(\phi) X$,
where
$$X = \left( \begin{array}{c} c t \\ x\end{array}\right)
\hspace{5ex} \textrm{and} \hspace{5ex} 
\Lambda(\phi) = \left( \begin{array}{cc}
\cosh \phi & -\sinh\phi \\
-\sinh\phi & \cosh \phi
\end{array}\right),$$
and where $\phi$ is the rapidity. 
Boost again along the $x$ direction, 
$X'' = \Lambda(\phi')X' = \Lambda(\phi')\Lambda(\phi) X$. 
Using the properties of the hyperbolic trigonometric functions, we can see that 
$X'' = \Lambda(\phi'') X$, 
where $\phi'' = \phi + \phi'$. 
(Do the matrix multiplication, $\Lambda(\phi')\Lambda(\phi)$.)
Thus, the composition of boosts along the $x$ direction is another boost along the $x$ direction.
The rest of the group axioms are also satisfied. 
Thus, the collection of boosts along the $x$ direction form a group. 
The group structure is independent of the representation. 
If the "representation" is not closed under composition, it is definitely not a representation of the group. 
Lorentz transformations can be represented by quaternions. 
See here for example. 
