The Hermitian Subspace under the Three Topologies

Introduction

The Hermitian subspace $\mathbb{M}_+$ is one of the six distinguished subspaces of the biquaternion algebra $\mathbb{B}$, defined and developed in Introduction to the Six Subspaces in the Algebra group. It reads the subspace for its basis, its defining involution, its algebra and module structure and its elements; this one reads it for its topology, and it does so three times.

The algebra carries three pairings of its elements — the bilinear form $B$, the Hermitian form $\langle\cdot,\cdot\rangle$ and the Krein form $[\cdot,\cdot]$ of The Three Pairings of the Biquaternion Algebra, built on the natural conjugation ${}^{\natural}$, the Hermitian conjugation ${}^{*}$ and the complex conjugation $\bar{\cdot}$. Each pairing restricts to the subspace, and each restriction is a form in its own right, with its own signature, its own definiteness, its own null set and its own group of isometries; each therefore induces its own topology on the subspace. The three are kept apart in three separate sections below, and they are compared in the table at the end.

Conventions. $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, the element $\tilde{Q}=\sum_{\mu}Q_{\mu}e_{\mu}$ with $Q_{\mu}=q_{\mu}+iq'_{\mu}$, units $e_0=1$ and $e_k^2=-e_0$, central scalar imaginary $i$, scalar part $\mathrm{Sc}$, sign vector $\varepsilon=(1,-1,-1,-1)$ and $E=\mathrm{diag}(1,-1,-1,-1)$.

The Subspace

Definition. The Hermitian subspace is the fixed space of Hermitian conjugation, $\mathbb{M}_+=\{\tilde{Q}:\tilde{Q}^{*}=\tilde{Q}\}$. It is the set of elements with real scalar part and purely imaginary vector part.

It is $\mathrm{span}_{\mathbb{R}}\{e_0,ie_1,ie_2,ie_3\}$, of real dimension $4$; it is not a subalgebra, and it is a Jordan algebra for the symmetrized product.

The Topology Induced by the Bilinear Form

Theorem (the restriction of the bilinear form). On $\mathbb{M}_+$ the bilinear form is

$$ B(\tilde{Q},\tilde{Q}) = N(\tilde{Q}) = q_0^2-(q'_1)^2-(q'_2)^2-(q'_3)^2 , $$

a real form of signature $(1,3)$ on the 4 real dimensions of the subspace.

Proof. On the subspace $Q_0=q_0$ and $Q_k=iq'_k$, so $N(\tilde{Q})=q_0^2-\sum_k(q'_k)^2$, one positive and three negative directions.

The restriction is Lorentzian, and it is the same form as the Krein restriction on this subspace; the Hermitian sector is one of the two subspaces on which the bilinear form is Lorentzian rather than split.

The null set. the cone $q_0^2=(q'_1)^2+(q'_2)^2+(q'_3)^2$, of real dimension $3$; its non-zero points are the non-nilpotent zero divisors of the subspace, the source of the non-pure family of the algebra.

The isometry group. The restriction is a real form of signature $(1,3)$, so its group of real-linear isometries on $\mathbb{M}_+$ is the orthogonal group $O(1,3)$; inside the ambient isometry group $O_4(\mathbb{C})$ of The Three Pairings of the Biquaternion Algebra the elements that preserve $\mathbb{M}_+$ form the corresponding subgroup.

The Topology Induced by the Hermitian Form

Theorem (the restriction of the Hermitian form). On $\mathbb{M}_+$ the Hermitian form is

$$ \langle\tilde{Q},\tilde{Q}\rangle = q_0^2+(q'_1)^2+(q'_2)^2+(q'_3)^2 , $$

of signature $(4,0)$.

Proof. $\langle\tilde{Q},\tilde{Q}\rangle=\sum_\mu|Q_\mu|^2=q_0^2+\sum_k|iq'_k|^2$.

The restriction is the Euclidean square on the four real coordinates, and it is the form that makes the subspace a Hilbert space; the Hermitian sector is the natural home of the positive definite structure.

The Euclidean topology. The restriction is positive definite, so it is a Euclidean inner product on the 4 real dimensions of the subspace. It defines the Euclidean norm $\lVert\tilde{Q}\rVert_E$, the distance and the balls, and hence the Euclidean topology of the subspace; on the algebra as a whole this is the topology of The Euclidean Topology of the Biquaternion Algebra. Its group of real-linear isometries on $\mathbb{M}_+$ is the compact orthogonal group $O(4)$, contained in the ambient unitary group $U(4)$ of The Unitary Group of the Biquaternion Algebra.

The null set. The form is positive definite, so $\langle\tilde{Q},\tilde{Q}\rangle=0$ holds only at $\tilde{Q}=0$: the subspace carries no isotropic vector for the Hermitian form.

The Topology Induced by the Krein Form

Theorem (the restriction of the Krein form). On $\mathbb{M}_+$ the Krein form is

$$ [\tilde{Q},\tilde{Q}] = q_0^2-(q'_1)^2-(q'_2)^2-(q'_3)^2 , $$

of signature $(1,3)$.

Proof. $[\tilde{Q},\tilde{Q}]=\sum_\mu\varepsilon_\mu|Q_\mu|^2=q_0^2-\sum_k(q'_k)^2$.

The Krein restriction coincides with the bilinear one on this subspace: the two indefinite forms agree here, and the Krein form is Lorentzian with the scalar direction timelike.

The null set. the same cone $q_0^2=\sum_k(q'_k)^2$ as for the bilinear topology: the null sets of the bilinear and the Krein forms coincide, since the two forms coincide.

The isometry group. The restriction is a real form of signature $(1,3)$, so its group of real-linear isometries on $\mathbb{M}_+$ is $O(1,3)$; inside the ambient group $U(1,3)$ of The Krein Isometry Group and Its $J$-Contractions the elements preserving $\mathbb{M}_+$ form the corresponding subgroup.

The Three Topologies Compared

The three restrictions are collected in one table; each entry is a form on the same real vector space $\mathbb{M}_+$, and the signatures are those of the underlying real form.

topology	form	restriction on $\mathbb{M}_+$	signature	definiteness	null set	isometry group
bilinear	$B=\mathrm{Sc}(\tilde{Q}^{\natural}\tilde{Q})$	$q_0^2-(q'_1)^2-(q'_2)^2-(q'_3)^2$	$(1,3)$	indefinite (Lorentzian)	$q_0^2=\sum_k(q'_k)^2$	$O(1,3)$
Hermitian	$\langle\tilde{Q},\tilde{Q}\rangle=\mathrm{Sc}(\tilde{Q}^{*}\tilde{Q})$	$q_0^2+(q'_1)^2+(q'_2)^2+(q'_3)^2$	$(4,0)$	positive definite	$\{0\}$	$O(4)$
Krein	$[\tilde{Q},\tilde{Q}]=\mathrm{Sc}(\bar{\tilde{Q}}\tilde{Q})$	$q_0^2-(q'_1)^2-(q'_2)^2-(q'_3)^2$	$(1,3)$	indefinite (Lorentzian)	$q_0^2=\sum_k(q'_k)^2$	$O(1,3)$

The bilinear and the Krein topologies coincide on this subspace — both Lorentzian, with the same cone of zero divisors — and the Hermitian topology is the positive definite one. This is the reverse of the centre, where it was the two definite forms that coincided.

Summary

On the Hermitian subspace the bilinear and the Krein forms coincide and are the Lorentzian form $q_0^2-\sum_k(q'_k)^2$ of signature $(1,3)$, with the three-dimensional cone $q_0^2=\sum_k(q'_k)^2$ as null set, and the Hermitian form is the Euclidean square $q_0^2+\sum_k(q'_k)^2$ of signature $(4,0)$, positive definite. The isometry groups are $O(1,3)$, $O(4)$ and $O(1,3)$.

Summary of Notation

Symbol	Meaning
$\mathbb{M}_+$	the Hermitian subspace, of real dimension $4$
$B$, $\langle\cdot,\cdot\rangle$, $[\cdot,\cdot]$	the bilinear, Hermitian and Krein forms of The Three Pairings of the Biquaternion Algebra
$(1,3)$	the signature of the bilinear form on $\mathbb{M}_+$
$(4,0)$	the signature of the Hermitian form on $\mathbb{M}_+$, positive definite
$(1,3)$	the signature of the Krein form on $\mathbb{M}_+$
$O(1,3)$, $O(4)$, $O(1,3)$	the isometry groups of the three restrictions