As the biquaternions have been a fixture of
linear algebra since the beginnings of
mathematical physics, there is an array of concepts that are illustrated or represented by biquaternion algebra. The
transformation group G = \lbrace g : g g^* = 1 \rbrace has two parts, G \cap H and G \cap M. The first part is characterized by g = \bar{g} ; then the Lorentz transformation corresponding to is given by T(q) = g^{-1} q g since g^* = g^{-1}. Such a transformation is a
rotation by quaternion multiplication, and the collection of them is \cong G \cap H . But this subgroup of is not a
normal subgroup, so no
quotient group can be formed. To view G \cap M it is necessary to show some subalgebra structure in the biquaternions. Let represent an element of the
sphere of square roots of minus one in the real quaternion subalgebra . Then and the plane of biquaternions given by D_r = \lbrace z = x + yhr : x, y \in \mathbb R \rbrace is a commutative subalgebra isomorphic to the plane of split-complex numbers. Just as the ordinary complex plane has a unit circle, D_r has a
unit hyperbola given by : \exp(ahr) = \cosh(a) + hr\ \sinh(a),\quad a \in R. Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because \exp(ahr) \exp(bhr) = \exp((a+b)hr). Hence these algebraic operators on the hyperbola are called
hyperbolic versors. The unit circle in and unit hyperbola in are examples of
one-parameter groups. For every square root of minus one in , there is a one-parameter group in the biquaternions given by G \cap D_r. The space of biquaternions has a natural
topology through the
Euclidean metric on -space. With respect to this topology, is a
topological group. Moreover, it has analytic structure making it a six-parameter
Lie group. Consider the subspace of bivectors A = \lbrace q : q^* = -q \rbrace . Then the
exponential map \exp:A \to G takes the real vectors to G \cap H and the -vectors to G \cap M. When equipped with the
commutator, forms the
Lie algebra of . Thus this study of a
six-dimensional space serves to introduce the general concepts of
Lie theory. When viewed in the matrix representation, is called the
special linear group in . Many of the concepts of
special relativity are illustrated through the biquaternion structures laid out. The subspace corresponds to
Minkowski space, with the four coordinates giving the time and space locations of events in a resting
frame of reference. Any hyperbolic versor corresponds to a
velocity in direction of speed where is the
velocity of light. The inertial frame of reference of this velocity can be made the resting frame by applying the
Lorentz boost given by since then g^{\star} = \exp(-0.5ahr) = g^* so that T(\exp(ahr)) = 1 . Naturally the
hyperboloid G \cap M, which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this "velocity space" with the
hyperboloid model of
hyperbolic geometry. In special relativity, the
hyperbolic angle parameter of a hyperbolic versor is called
rapidity. Thus we see the biquaternion group provides a
group representation for the
Lorentz group. After the introduction of
spinor theory, particularly in the hands of
Wolfgang Pauli and
Élie Cartan, the biquaternion representation of the Lorentz group was superseded. The new methods were founded on
basis vectors in the set : \{ q \ :\ q q^* = 0 \} = \left\{ w + x\mathbf i + y\mathbf j + z\mathbf k \ :\ w^2 + x^2 + y^2 + z^2 = 0 \right\} which is called the
complex light cone. The above
representation of the Lorentz group coincides with what physicists refer to as
four-vectors. Beyond four-vectors, the
Standard Model of particle physics also includes other Lorentz representations, known as
scalars, and the -representation associated with e.g. the
electromagnetic field tensor. Furthermore, particle physics makes use of the representations (or
projective representations of the Lorentz group) known as left- and right-handed
Weyl spinors,
Majorana spinors, and
Dirac spinors. It is known that each of these seven representations can be constructed as invariant subspaces within the biquaternions. == As a composition algebra ==