The Anti-Hermitian Subspace under the Three Topologies

Introduction

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

It is $\mathrm{span}_{\mathbb{R}}\{ie_0,e_1,e_2,e_3\}$, of real dimension $4$; it is not a subalgebra, and it is a Lie algebra for the commutator.

The Topology Induced by the Bilinear Form

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

$$ B(\tilde{Q},\tilde{Q}) = N(\tilde{Q}) = (q_1^2+q_2^2+q_3^2)-(q'_0)^2 , $$

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

Proof. On the subspace $Q_0=iq'_0$ and $Q_k=q_k$, so $N(\tilde{Q})=(q_1^2+q_2^2+q_3^2)-(q'_0)^2$, three positive and one negative direction.

The restriction is the negative of the Hermitian restriction, of signature $(3,1)$ rather than $(1,3)$; the two sectors are exchanged by multiplication by $i$.

The null set. the cone $\sum_kq_k^2=(q'_0)^2$, of real dimension $3$, the zero divisors of the subspace, none of which is nilpotent.

The isometry group. The restriction is a real form of signature $(3,1)$, so its group of real-linear isometries on $\mathbb{M}_-$ is the orthogonal group $O(3,1)$; 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. $\langle\tilde{Q},\tilde{Q}\rangle=\sum_\mu|Q_\mu|^2=|iq'_0|^2+\sum_kq_k^2$.

The restriction is the Euclidean square on the four real coordinates, positive definite, exactly as on the Hermitian sector.

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. $[\tilde{Q},\tilde{Q}]=\sum_\mu\varepsilon_\mu|Q_\mu|^2=(q'_0)^2-\sum_kq_k^2$, the signs $\varepsilon_k=-1$ falling on the real vector directions.

The Krein restriction is the negative of the bilinear restriction, of signature $(1,3)$ rather than $(3,1)$; the passage between them is again the sign vector $\varepsilon$.

The null set. the cone $(q'_0)^2=\sum_kq_k^2$, of real dimension $3$, the light cone of the Lorentzian form on the subspace.

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	$B=\mathrm{Sc}(\tilde{Q}^{\natural}\tilde{Q})$	$(q_1^2+q_2^2+q_3^2)-(q'_0)^2$	$(3,1)$	indefinite (Lorentzian)	$\sum_kq_k^2=(q'_0)^2$	$O(3,1)$
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)	$(q'_0)^2=\sum_kq_k^2$	$O(1,3)$

The two indefinite topologies are negatives of one another — signatures $(3,1)$ for the bilinear form and $(1,3)$ for the Krein form — and the Hermitian topology is positive definite. The subspace is the negative counterpart of the Hermitian sector in all three topologies at once.

Summary

On the anti-Hermitian subspace the bilinear form restricts to $\sum_kq_k^2-(q'_0)^2$ of signature $(3,1)$, with the cone $\sum_kq_k^2=(q'_0)^2$ as null set; the Hermitian form to the Euclidean square $(q'_0)^2+\sum_kq_k^2$ of signature $(4,0)$, positive definite; and the Krein form to $(q'_0)^2-\sum_kq_k^2$ of signature $(1,3)$, the negative of the bilinear restriction. The isometry groups are $O(3,1)$, $O(4)$ and $O(1,3)$.

Summary of Notation

Symbol	Meaning
$\mathbb{M}_-$	the anti-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
$(3,1)$	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 $\mathbb{M}_-$
$O(3,1)$, $O(4)$, $O(1,3)$	the isometry groups of the three restrictions