The Unit Group and the Frobenius Norm in the Matrix Representation

Introduction

The Euclidean norm of the biquaternion algebra is the Frobenius norm of its matrix realization, and the two topological objects attached to the algebra — its underlying eight-dimensional real space and its group of units — are therefore read most directly in the matrix picture. This article collects those readings. It is the TOPOLOGY entry of the matrix-representation group. The general topology of the algebra is owned by Biquaternion Topology and the general group topology by The Biquaternion Unit Group as a Topological Group; nothing of either is reproved here, and the article states only what the realization $\Phi$ adds.

The realization is that of Biquaternion 2×2 Matrix Element Representation, with $\Phi(e_0) = I$ and $\Phi(e_k) = -i\sigma_k$; the Euclidean norm and the Frobenius norm are those of The Forms in the Matrix Representation of the Biquaternion Algebra.

The Frobenius Norm and the Isometry

Proposition (the realization is a similarity of Euclidean spaces). For every biquaternion $\tilde{Q}$,

$$ \bigl\|\Phi(\tilde{Q})\bigr\|_F = \sqrt 2 \,\bigl\|\tilde{Q}\bigr\|_E , $$

so the realization, regarded as a real-linear map

$$ \Phi : \mathbb{B}_{\mathbb{R}} \longrightarrow M_2(\mathbb{C})_{\mathbb{R}} \cong \mathbb{R}^8 , $$

is a linear isometry up to the fixed scale $\sqrt 2$: it is injective, it is onto, and it multiplies all lengths by the same constant.

Proof. The trace identity of the companion FORM article gives $\|\Phi(\tilde{Q})\|_F^2 = \operatorname{Tr}(\tilde{Q}\tilde{Q}^{*}) = 2\|\tilde{Q}\|_E^2$, and the square roots agree. Injectivity and surjectivity are the isomorphism of the realization. $\square$

Two immediate consequences, recorded here because they are read from the realization:

The underlying topology of the algebra is the Euclidean topology of $\mathbb{R}^8$, transported along $\Phi$; the algebra is a complete, contractible real space, and its unit sphere in the Euclidean norm is the $7$-sphere $S^7$. The development of the underlying space and its contractibility is The Euclidean Topology of the Biquaternion Algebra, §The Contractibility of the Algebra.
The scale $\sqrt 2$ is the trace normalisation $\operatorname{Tr}\Phi(\tilde{Q}) = 2Q_0$; with the normalised map $\Phi/\sqrt 2$ the realization becomes an exact isometry between $\mathbb{B}$ with the Euclidean norm and $M_2(\mathbb{C})$ with the Frobenius norm.

The Unit Group

Proposition (the unit groups in matrix form). Under $\Phi$,

$$ \mathbb{B}^\times \cong GL_2(\mathbb{C}), \qquad \{\tilde{Q} : N(\tilde{Q}) = 1\} \cong SL_2(\mathbb{C}). $$

Proof. The realization is an algebra isomorphism, so it carries invertible elements to invertible matrices, giving the first statement; and it carries the biquaternion norm to the determinant, $N(\tilde{Q}) = \det\Phi(\tilde{Q})$, so the locus $N = 1$ is the determinant-one locus, giving the second. $\square$

The unit-norm group is the group that acts on the Hermitian subspace in the two-sided action of Biquaternion 4×4 Regular Matrix Element Representation, §The Two-Sided Action; in the matrix picture it is $SL_2(\mathbb{C})$ acting by $*$-congruence on the Hermitian matrices.

The Maximal Compact Slice

Proposition (the compact slice). The maximal compact subgroup of $\mathbb{B}^\times \cong GL_2(\mathbb{C})$ is $U(2)$, and the maximal compact subgroup of the unit-norm group $\cong SL_2(\mathbb{C})$ is $SU(2)$, which is the double cover of $SO(3)$:

$$ SU(2) \cong \mathrm{Spin}(3), \qquad SU(2)/\{\pm I\} \cong SO(3). $$

Proof. The unitary matrices are the compact subgroup of the complex general linear group by the polar decomposition, and the determinant-one unitary matrices are the compact subgroup of the special linear group; the double cover of $SO(3)$ by $SU(2)$ is the standard identification of the unit quaternions. $\square$

The slice is the matrix form of the maximal compact subgroup of the unit group, whose retraction and homotopy are owned by The Unitary Group of the Biquaternion Algebra, §The Structure of the Unitary Group; here it is recorded only as the unitary slice of the realization.

The Unit Spheres

The Euclidean unit spheres of the algebra and of its distinguished subspaces have a direct matrix reading.

object	Euclidean description	matrix description
the algebra $\mathbb{B}$	unit sphere $S^7$	the Frobenius sphere $\\|\Phi(\tilde{Q})\\|_F^2 = 2$
the quaternion subspace $\mathbb{H}_{\mathbb{B}}$	unit sphere $S^3$	the Frobenius sphere on the complex matrices with real entries
the roots of $-1$ in $\mathbb{H}_{\mathbb{B}}$	$S^2$	the traceless anti-Hermitian matrices of Frobenius norm $\sqrt 2$
the Hermitian subspace $\mathbb{M}_+$	the null cone and its link	the Hermitian matrices, of signature $(1,3)$

The third row is the matrix form of the root sphere of The Six Subspaces and the Roots of Minus One and of the classification of Biquaternion Square Roots of Minus One, Zero and Plus One; the general topology of the unit sphere is The Euclidean Topology of the Biquaternion Algebra, §The Euclidean Unit Sphere, and that of the link of the null cone is Biquaternion Topology, §The link of the null cone.

The Lorentz Group and Its Connectedness

The one topological statement that the realization adds to the units is the connectedness of the Lorentz image.

Proposition (the image is connected). The two-sided action of the unit-norm group on $\mathbb{M}_+$,

$$ SL_2(\mathbb{C}) \times \mathbb{M}_+ \longrightarrow \mathbb{M}_+, \qquad (\tilde{A}, \tilde{Q}) \longmapsto \tilde{A}\tilde{Q}\tilde{A}^{*}, $$

has image in the identity component $SO^+(1,3)$ of the orthogonal group of the form, because $SL_2(\mathbb{C})$ is connected and the action is continuous.

Proof. The action is continuous in $\tilde{A}$, the group $SL_2(\mathbb{C})$ is connected as a complex algebraic group, and the continuous image of a connected set is connected; the image therefore lies in the identity component of the orthogonal group. The double cover itself, with its central kernel of order two, is Biquaternion 4×4 Regular Matrix Element Representation, §The Two-Sided Action; the group and its geometry are Biquaternion Rotations and Lorentz Transformations. $\square$

Summary

The matrix realization is a linear similarity of Euclidean spaces, $\|\Phi(\tilde{Q})\|_F = \sqrt2\|\tilde{Q}\|_E$, so the topology of the algebra is the Euclidean topology of $\mathbb{R}^8$ read on the matrices, with unit sphere $S^7$ and its subspaces carrying $S^3$ and $S^2$ in the familiar places. The unit group is $GL_2(\mathbb{C})$ and the unit-norm group is $SL_2(\mathbb{C})$, because $N = \det$; their maximal compact subgroups are $U(2)$ and $SU(2) = \mathrm{Spin}(3)$. The two-sided action of the unit-norm group on the Hermitian matrices has image in $SO^+(1,3)$, and its connectedness is the connectedness of $SL_2(\mathbb{C})$. The general topology of the algebra and of its unit group is Biquaternion Topology and The Biquaternion Unit Group as a Topological Group; this article is their matrix reading.

Summary of Notation

Symbol	Meaning
$\\|\cdot\\|_F$	Frobenius norm; $\\|\Phi(\tilde{Q})\\|_F = \sqrt2\,\\|\tilde{Q}\\|_E$
$S^7$	Euclidean unit sphere of $\mathbb{B}_{\mathbb{R}} \cong \mathbb{R}^8$
$S^3$, $S^2$	Unit spheres of the quaternion subspace and of its root set
$GL_2(\mathbb{C})$	Group of units of the realization
$SL_2(\mathbb{C})$	Unit-norm group, $\{\tilde{Q} : N(\tilde{Q}) = 1\}$
$U(2)$, $SU(2)$	Maximal compact subgroups of $GL_2(\mathbb{C})$ and $SL_2(\mathbb{C})$
$\mathrm{Spin}(3) \cong SU(2)$	Double cover of $SO(3)$
$SO^+(1,3)$	Identity component of the orthogonal group, image of the two-sided action