History
In 1891, Alexander Macfarlane proposed a variant of Hamilton’s quaternions which appeared much the same but whose components i, j, and k squared to identity rather than its negative. Such a simple twist led to a radically different structure which lacked a whole host of the usual nicely behaved properties (most notably associativity). This particular algebra is my topic of focus, and while it might on the surface appear unruly and antiquated, the pathologies may yet become the source of richness in their defects.
Definition and multiplication table
A hyperbolic quaternion is a linear combination of the elements \(\lbrace 1, i, j, k \rbrace\) using real coefficients such that \[q = q^0 1 + q^1 i + q^2 j + q^3 k, \quad q^0, q^1, q^2, q^3 \in \Re. \] Multiplication is defined according to the following rules: \[i^2 = j^2 = k^2 = 1, \\ ij = -ji = k \\ jk=-kj = i, \\ ki = -ik = j \] The multiplication table is then \[ \begin{array}{|c|c|c|c|c|} \hline \bf{x} & \bf{1} & \bf{i} & \bf{j} & \bf{k} \\ \hline \bf{1} & \phantom{-}1\phantom{-} & \phantom{-}i\phantom{-} & \phantom{-}j\phantom{-} & \phantom{-}k\phantom{-} \\ \hline \bf{i} & \phantom{-}i\phantom{-} & \phantom{-}1\phantom{-} & \phantom{-}k\phantom{-} & -j\phantom{-} \\ \hline \bf{j} & \phantom{-}j\phantom{-} & -k\phantom{-} & \phantom{-}1\phantom{-} & \phantom{-}i\phantom{-} \\ \hline \bf{k} & \phantom{-}k\phantom{-} & \phantom{-}j\phantom{-} & -i\phantom{-} & \phantom{-}1\phantom{-} \\ \hline \end{array} \]
Conjugate and Quadratic Form
The conjugate of a hyperbolic quaternion \(q = q^0 1 + q^1 i + q^2 j + q^3 k\) is \(q^* = q^0 1 – q^1 i – q^2 j – q^3 k\), and thus the product of q with its conjugate is thus \[q(q^*) = (q^0)^2 – (q^1)^2 – (q^2)^2 – (q^3)^2 \] This matches the quadratic form used in spacetime theory.
Properties
Much like Hamilton’s quaternions, these hyperbolic ones are non-commutative: \[ij=-ji\] Additionally, Macfarlane’s algebra is non-alternative and, consequently, non-associative. The failure of the associative law is demonstrated by a simple calculation: \[(ii)j=j, \quad i(ij)=ik=-j\] Since \((ii)j \neq i(ij)\), the algebra is non-associative. This specific pathology directly causes the matrix representations of left multiplication, \(M^L_a\), to not be closed under matrix multiplication (a crucial distinction from associative algebras).
The basis elements {i, j, k} are also flexible: \[(ij)i=i(ji)=j \\ (ik)i=i(ki)=k \\ (ji)j=j(ij)=i \\ (jk)j=j(kj)=k \\ (ki)k=k(ik)=i \\ (kj)k=k(jk)=j\]The algebra possesses idempotents and zero-divisors: \[{\left( \frac{1 \pm i}{2} \right)}^2 = \left(\frac{1 \pm i}{2}\right) \\ \left(\frac{1 + i}{2} \right) \left(\frac{1-i}{2}\right)=0\]
Matrix representation of left/Right Multiplication
The hyperbolic quaternions, being a non-associative algebra (a key feature of Macfarlane’s 1891 construction), cannot be represented by matrices in a way that preserves full algebraic structure (i.e., the mapping \(q \mapsto M^L_q\) is not a ring homomorphism). As a result, the linear transformations for multiplication are not closed under matrix multiplication, meaning \((M^L_i)^2 \neq M^L_{i^2}\).
However, left and right multiplication by \(i, j, k\) can still be represented as a linear transformation acting on the vector space \(\mathbb{R}^4\). These matrices are derived from the algebra’s structure constants.
\[M_i^L = \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix} \quad M_i^R = \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\end{pmatrix}\]
\[M_j^L = \begin{pmatrix} 0&0&1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\end{pmatrix} \quad M_j^R = \begin{pmatrix} 0&0&1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{pmatrix}\]
\[M_k^L = \begin{pmatrix} 0&0&0&1\\0&0&-1&0\\0&1&0&0\\1&0&0&0\end{pmatrix} \quad M_k^R = \begin{pmatrix} 0&0&0&1\\0&0&1&0\\0&-1&0&0\\1&0&0&0\end{pmatrix}\]
Lorentz Group Generator Basis
The connection between the algebra’s multiplication operators and the structure of \(4D\) spacetime becomes clear when we look at the generators of the Lorentz Group, \(\mathrm{SO}(1,3)\), which preserves the spacetime metric. The six matrices \(\{K_a, J_a\}\) form a basis for the Lorentz Lie algebra, \(\mathfrak{so}(1,3)\).
The real 4×4 block matrix boost \(K_1, K_2, K_3\) and rotation \(J_1, J_2, J_3\) generator matrices are
\[K_1 = \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix} \quad J_1 = \begin{pmatrix} 0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix}\]
\[K_2 = \begin{pmatrix} 0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\end{pmatrix} \quad J_2 = \begin{pmatrix} 0&0&0&0\\0&0&0&1\\0&0&0&0\\0&-1&0&0\end{pmatrix}\]
\[K_3 = \begin{pmatrix} 0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{pmatrix} \quad J_3 = \begin{pmatrix} 0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{pmatrix}\]
Thus, one can see that for the hyperbolic quaternion left and right multiplication matrices \(M^L_a\) and \(M^R_a\) where \(a \in \lbrace i/1, j/2, k/3 \rbrace \): \[M^L_a = K_a + J_a\] \[M^R_a = K_a – J_a\]
This correspondence illuminates how the linear transformations representing left and right multiplication by the basis elements \(i, j, k\) provide a direct basis for the Lie algebra \(\mathfrak{so}(1,3)\). Specifically, these transformations can be seen as linear combinations of the Lorentz boosts (\(K_a\)) and spatial rotations (\(J_a\)), thereby connecting Macfarlane’s unconventional algebra directly to the symmetries of Minkowski spacetime. This observation suggests a potential utility for the hyperbolic quaternions in representing Lorentz transformations.
Commutators
Commutators K boosts and J rotations take the usual form, accompanied by their hyperbolic quaternion matrix counterparts: \[ [J_a, J_b] = \epsilon_{abc} J_c = \frac{1}{2} \epsilon_{abc} (M_c^L – M_c^R), \] \[[J_a, K_b] = \epsilon_{abc} K_c = \frac{1}{2} \epsilon_{abc} (M_c^L + M_c^R), \] \[[K_a, K_b] = -\epsilon_{abc} J_c = -\frac{1}{2} \epsilon_{abc} (M_c^L – M_c^R). \]
Commutators of the left and right multiplication matrices \(M^L_a\) and \(M^R_a\) are as follows: \[ [M^L_a, M^L_b] = \epsilon_{abc} (M^L_c + M^R_c) = 2 \epsilon_{abc} K_c, \] \[ [M^L_a, M^R_b] = -\epsilon_{abc} (M^L_c – M^R_c) = -2 \epsilon_{abc} J_c, \] \[ [M^R_a, M^R_b] = – \epsilon_{abc} (M^L_c + M^R_c) = -2 \epsilon_{abc} K_c. \]
In Summary
The hyperbolic quaternions are similar in definition to Hamilton’s but for their elements squaring to positive identity. This leads to a failure of certain properties (associativity and alternativity) and results in the existence of idempotents and zero-divisors. Left and right multiplication by i, j, k (as represented by linear transformations) provide a direct basis for the generators of the Lorentz Lie algebra \(\mathfrak{so}(1,3)\) (\(K_a\) and \(J_a\)). In future posts, other implications and applications will be discussed regarding these particular properties.
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