Penumbra's Notebook

Definition Reminder

A hyperbolic quaternion (Macfarlane non-associative variety) 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: \[i^2 = j^2 = k^2 = 1, \\ ij = -ji = k \\ jk=-kj = i, \\ ki = -ik = j \]

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Matrix Representations

The matrix representations of left or right multiplication by i, j, k are:

\[M_i^L = \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix} \quad M_i^R = \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\end{pmatrix}\]

\[M_j^L = \begin{pmatrix} 0&0&1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\end{pmatrix} \quad M_j^R = \begin{pmatrix} 0&0&1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{pmatrix}\]

\[M_k^L = \begin{pmatrix} 0&0&0&1\\0&0&-1&0\\0&1&0&0\\1&0&0&0\end{pmatrix} \quad M_k^R = \begin{pmatrix} 0&0&0&1\\0&0&1&0\\0&-1&0&0\\1&0&0&0\end{pmatrix}\]

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Closure of the Six Matrices

While the algebra itself is closed under multiplication, the representations of its left and right multiplicative action as linear transformations are not closed. Repeated application of the 6 matrices inevitably results in 64 matrices total:

\[ \left(\mathbb{Z}_2 \right)^4 \rtimes V_4 \]

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Permutation of Basis

The closure does not depend on assigning the basis to be \(\lbrace 1, i, j, k \rbrace\). Permutations of the assignment of basis (such as \(\lbrace j,k,1,i \rbrace\)) will still give 6 matrices for left/right multiplication by i, j, k which while different from those in another basis will still give the same 64-element closure under repeated multiplication.

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Hamilton Quaternion Subgroup

Real 4×4 matrix representations of Hamilton’s quaternions form an associative subgroup of the 64-element closure.

\[M_i^L = \begin{pmatrix} 0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix} \quad M_i^R = \begin{pmatrix} 0&-1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\end{pmatrix}\]

\[M_j^L = \begin{pmatrix} 0&0&-1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\end{pmatrix} \quad M_j^R = \begin{pmatrix} 0&0&-1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{pmatrix}\]

\[M_k^L = \begin{pmatrix} 0&0&0&-1\\0&0&-1&0\\0&1&0&0\\1&0&0&0\end{pmatrix} \quad M_k^R = \begin{pmatrix} 0&0&0&-1\\0&0&1&0\\0&-1&0&0\\1&0&0&0\end{pmatrix}\]