Remarkable Subspaces under the General Quaternionic Algebra
Introduction
The general quaternionic bilinear form $\langle\tilde Q_1,\tilde Q_2\rangle_{\natural}=\sum_\mu Q_{1\mu} Q_{2\mu}$, whose diagonal is the biquaternion norm, is read here on the remarkable real subspaces of Introduction to the Remarkable Subspaces – the centre, the vector subspace, the quaternion subspace, the anti-quaternion subspace, the Hermitian subspace and the anti-Hermitian subspace. On each, separating the real and imaginary parts of the coefficients turns the restriction into a real symmetric form, and this article records the signatures, the restriction Gram matrices, the two definite rows and the maximal definite subspaces, the isotropic lines of each indefinite restriction, and the automorphism group of each restriction.
This article is the development of the restriction table stated in The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra; the form itself, its polarisation and its quadratic space are in Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra. The signatures are recorded in §The Signatures and the Restriction Matrices; the remarkable subspaces are Introduction to the Remarkable Subspaces; the other three forms on the same remarkable subspaces are Remarkable Subspaces and the Four General Products, Forms and Norms, Remarkable Subspaces under the General Plain Sesqualgebra and Remarkable Subspaces under the General Quaternionic Sesqualgebra; and the null structure of the form on the subspaces is The Isotropic Structure of the General Quaternionic Algebra.
The Signatures and the Restriction Matrices
The restriction of the form to a subspace is the real symmetric form obtained by taking the real part on the real span; because the form is $\mathbb{C}$-bilinear and $i$ is central, the cross terms between a real direction and its imaginary companion are purely imaginary and drop out, so the restricted form is diagonal in the natural real basis.
| Subspace | Real basis | Restricted norm | Gram matrix | Signature |
|---|---|---|---|---|
| Centre $\mathbb{C}_{\mathbb{B}}$ | $e_t,\,ie_t$ | $q_t^2-(q'_t)^2$ | $\operatorname{diag}(1,-1)$ | $(1,1)$ |
| Vector $\mathrm{Vect}(\mathbb{B})$ | $\vec e_k,\,i\vec e_k$ | $\sum_k\bigl(q_k^2-(q'_k)^2\bigr)$ | $\operatorname{diag}(I_3,-I_3)$ | $(3,3)$ |
| Quaternion $\mathbb{H}_{\mathbb{B}}$ | $e_t,\vec e_x,\vec e_y,\vec e_z$ | $\sum_\mu q_\mu^2$ | $I_4$ | $(4,0)$ |
| Anti-quaternion $i\mathbb{H}_{\mathbb{B}}$ | $ie_t,i\vec e_x,i\vec e_y,i\vec e_z$ | $-\sum_\mu(q'_\mu)^2$ | $-I_4$ | $(0,4)$ |
| Hermitian $\mathbb{M}_+$ | $e_t,i\vec e_x,i\vec e_y,i\vec e_z$ | $q_t^2-\sum_k(q'_k)^2$ | $\operatorname{diag}(1,-1,-1,-1)$ | $(1,3)$ |
| Anti-Hermitian $\mathbb{M}_-$ | $ie_t,\vec e_x,\vec e_y,\vec e_z$ | $\sum_kq_k^2-(q'_t)^2$ | $\operatorname{diag}(-1,1,1,1)$ | $(3,1)$ |
Proposition (the matrices). In the natural real basis displayed, the Gram matrix of the restricted form is the one tabulated; each restriction is non-degenerate, and its determinant is $\pm1$.
Proof. The entries are $\langle u,v\rangle_{\natural}=\mathrm{Sc}(uv^{\natural})$ for $u,v$ among the real basis of the subspace; a real basis element is $e_\mu$ or $ie_\mu$, and $\mathrm{Sc}(e_\mu e_\nu^{\natural})=\delta_{\mu\nu}$, $\mathrm{Sc}(e_\mu(ie_\nu)^{\natural})=-i\delta_{\mu\nu}$, $\mathrm{Sc}(ie_\mu e_\nu^{\natural})=i\delta_{\mu\nu}$ and $\mathrm{Sc}(ie_\mu(ie_\nu)^{\natural})=-\delta_{\mu\nu}$. The cross entries are purely imaginary, hence zero in the real part, and the two within-block entries give the displayed diagonal. Each matrix is invertible, of determinant $\pm1$.
Remark (the index and the defect of each restriction). The index of a restriction is the number of negative entries of its signature, and the defect or radical is the subspace orthogonal to the whole restriction. The restrictions are non-degenerate, their Gram matrices having determinant $\pm1$, so every defect is trivial; the six indexes are $1$, $3$, $0$, $4$, $3$ and $1$, read from the signatures $(1,1)$, $(3,3)$, $(4,0)$, $(0,4)$, $(1,3)$ and $(3,1)$. The index of the whole realified form on $\mathbb{R}^8$ is $4$, of signature $(4,4)$, and its defect is trivial too; that whole-form reading is in The Isotropic Structure of the General Quaternionic Algebra.
Remark (why the cross terms drop). The vanishing of the cross block is not a numerical accident: it is the statement that a real direction and its imaginary companion are orthogonal for the realified form, while the form takes the value $i$ between them before the real part is taken. The same computation gives the realification of the whole form, $\operatorname{diag}(I_4,-I_4)$ of signature $(4,4)$, of which the tables are the restrictions to the displayed subspaces.
Remark (the realified matrix in the two orderings). The realification is worth displaying once in each of its two natural orderings. In the grouped basis $e_t,\vec e_x,\vec e_y,\vec e_z,ie_t,i\vec e_x,i\vec e_y,i\vec e_z$, the four real directions first, the matrix is the block diagonal $\operatorname{diag}(I_4,-I_4)$; in the interleaved basis $e_t,ie_t,\vec e_x,i\vec e_x,\vec e_y,i\vec e_y,\vec e_z,i\vec e_z$, each complex direction kept beside its companion, it is $\operatorname{diag}(1,-1,1,-1,1,-1,1,-1)$, and the two differ only by the permutation that carries one ordering to the other. The four blocks of the interleaved reading are the centre and the three vector directions, and the remarkable subspaces of this article are cut from them in three ways: one block at a time, which gives the centre; all four blocks together, which gives the vector subspace; and the real or the imaginary half of all four, which gives the quaternion and anti-quaternion subspaces, with the two Hermitian subspaces obtained by exchanging the half of the centre block alone. Every restriction Gram matrix of the table is a submatrix of this diagonal matrix, in the order of signs displayed.
The Definite Rows and the Maximal Definite Subspaces
The quaternion subspace carries the signature $(4,0)$ and the anti-quaternion subspace the signature $(0,4)$: these are the definite rows, exchanged by multiplication by $i$, under which the norm changes sign,
$$ \boxed{\; \langle i\tilde Q_1,i\tilde Q_2\rangle_{\natural}=-\langle\tilde Q_1,\tilde Q_2\rangle_{\natural} \;}. $$
The quaternion subspace is thus a maximal positive definite subspace of the norm, of dimension $4$, and the anti-quaternion subspace a maximal negative definite one; they are orthogonal complements for the realified form, $\mathbb{H}_{\mathbb B}\oplus i\mathbb H_{\mathbb B}=\mathbb B$, since the real part of the pairing of a real direction with its imaginary companion vanishes, while the complex pairing itself takes the value $i$ between them; and the norm is the positive form on the first and its negative on the second. This splitting is the one the norm uses to decide invertibility: on a definite subspace the restriction vanishes only at the origin, so no nonzero element of either quaternion subspace is a zero divisor, and the classical quaternion algebra $\mathbb{H}_{\mathbb B}$ is a division algebra.
Proposition (the maximal definite dimensions). The maximal dimension of a positive definite subspace of the realified form is $4$, attained by $\mathbb{H}_{\mathbb B}$, and the maximal dimension of a negative definite subspace is $4$, attained by $i\mathbb H_{\mathbb B}$; inside the four indefinite rows the maximal definite dimensions are $1$ on the centre, $3$ on the vector subspace, $1$ (positive) and $3$ (negative) on the Hermitian subspace and $3$ (positive) and $1$ (negative) on the anti-Hermitian one.
Proof. Each bound is the smaller of the two inertia indices of the restricted form, read from its signature; the bound is attained by the span of the basis elements carrying the corresponding signs, which is definite because the restricted Gram matrix is diagonal in that basis. For the whole algebra the two bounds are the two dimensions of the definite rows, which add to the total dimension $8$ and are complementary.
The Isotropic Lines of the Indefinite Restrictions
Three of the restrictions are indefinite with an isotropic cone; the centre is indefinite but anisotropic over $\mathbb{C}$; the two definite rows have no isotropic element.
The vector subspace. The restriction is the complex quadratic $\sum_kQ_k^2$ on $\mathbb{C}\{\vec e_x,\vec e_y,\vec e_z\}$, the pure zero-divisor cone of Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors. Its isotropic complex lines are the points of the conic $\{[Q]\in\mathbb{P}^2:\sum_kQ_k^2=0\}$; the line $\mathbb{C}(\vec e_x+i\vec e_y)$ is one of them, and the maximal totally isotropic dimension is $1$ over $\mathbb{C}$ and $3$ over $\mathbb{R}$, the realification having signature $(3,3)$ and the real $3$-space $\mathbb{R}\{\vec e_x+i\vec e_x,\vec e_y+i\vec e_y,\vec e_z+i\vec e_z\}$ being totally isotropic. The cone has real dimension $4$.
The Hermitian and anti-Hermitian subspaces. The restrictions are the two real forms of the interval, of signatures $(1,3)$ and $(3,1)$; their null sets are the two real null cones, of real dimension $3$, and the isotropic real lines of a null cone are parametrised by the real unit directions. The maximal totally isotropic dimension is $1$ in each, a null line; the non-pure zero divisors are the elements of these two cones.
The centre. The restriction is the hyperbolic plane $q_t^2-(q'_t)^2$, so the realified restriction vanishes on the two real lines $\mathbb{R}(e_t+ie_t)$ and $\mathbb{R}(e_t-ie_t)$; the general quaternionic bilinear form $\langle\cdot,\cdot\rangle_{\natural}$ itself, however, reads $Q_tR_t$ on the complex line $\mathbb{C}e_t$, which is anisotropic, so the centre has no isotropic line for $\langle\cdot,\cdot\rangle_{\natural}$ and its null set is $\{0\}$. The two statements are the two readings of the same restriction, the realified one and the complex one, and they must not be confused.
The isotropic-line count. Intersected with the remarkable subspaces, the null cone has the real dimensions $0$, $4$, $0$, $0$, $3$ and $3$ in the order of the table, being empty off the origin on the two definite rows.
| Subspace | Null set of the restriction | Real dimension | Isotropic lines | Max totally isotropic |
|---|---|---|---|---|
| Centre $\mathbb{C}_{\mathbb B}$ | $\{0\}$ for $\langle\cdot,\cdot\rangle_{\natural}$; two lines for the realified form | $0$ | none for $\langle\cdot,\cdot\rangle_{\natural}$ | $0$ |
| Vector $\mathrm{Vect}(\mathbb B)$ | the complex cone $\sum_kQ_k^2=0$ | $4$ | the conic in $\mathbb{P}^2$ | $1$ over $\mathbb{C}$, $3$ over $\mathbb{R}$ |
| Quaternion $\mathbb H_{\mathbb B}$ | $\{0\}$ | $0$ | none | $0$ |
| Anti-quaternion $i\mathbb H_{\mathbb B}$ | $\{0\}$ | $0$ | none | $0$ |
| Hermitian $\mathbb M_+$ | the real null cone | $3$ | $S^2$ of null directions | $1$ |
| Anti-Hermitian $\mathbb M_-$ | the real null cone | $3$ | $S^2$ of null directions | $1$ |
The Automorphism Group of Each Restriction
Proposition. The automorphism group of the restriction of the form to a subspace is the real orthogonal group of its signature,
$$ \boxed{\; O(1,1),\quad O(3,3),\quad O(4),\quad O(4),\quad O(1,3),\quad O(3,1) \;} $$
on the centre, the vector subspace, the quaternion subspace, the anti-quaternion subspace, the Hermitian subspace and the anti-Hermitian subspace. Their real dimensions are $1$, $15$, $6$, $6$, $6$, $6$; the two definite ones are the quaternion and anti-quaternion rows, and the two real forms give the isomorphic groups $O(1,3)$ and $O(3,1)$, the two real forms of the same complex group.
Proof. A real-linear automorphism of a non-degenerate real symmetric form of signature $(n,m)$ is an element of $O(n,m)$; the dimension of $O(n,m)$ is $n(n-1)/2$ with $n=p+q$, giving $1$, $15$ and $6$; and the restrictions are non-degenerate by the determinant computation above. The definite rows give the groups $O(4)$; the two real forms give the isomorphic groups $O(1,3)$ and $O(3,1)$.
| Subspace | Restriction | Automorphism group | Real dimension | Definite |
|---|---|---|---|---|
| Centre $\mathbb{C}_{\mathbb B}$ | $(1,1)$ | $O(1,1)$ | $1$ | no |
| Vector $\mathrm{Vect}(\mathbb B)$ | $(3,3)$ | $O(3,3)$ | $15$ | no |
| Quaternion $\mathbb H_{\mathbb B}$ | $(4,0)$ | $O(4)$ | $6$ | yes |
| Anti-quaternion $i\mathbb H_{\mathbb B}$ | $(0,4)$ | $O(4)$ | $6$ | yes |
| Hermitian $\mathbb M_+$ | $(1,3)$ | $O(1,3)$ | $6$ | no |
| Anti-Hermitian $\mathbb M_-$ | $(3,1)$ | $O(3,1)$ | $6$ | no |
Remark (the restriction group is not induced by the ambient group). The automorphism groups of the table are the largest groups preserving the restricted form, and they are not the restrictions of the automorphism group of the ambient form: an automorphism of a subspace need not extend to the algebra, and conversely an automorphism of the whole form need not preserve a subspace. The ambient group is the complex orthogonal group $O_4(\mathbb{C})$, of complex dimension $6$ and real dimension $12$ (Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Quaternionic Norm); the group of real-linear automorphisms of the realified form is the larger $O(4,4)$, of real dimension $28$, and is not the subject of this article. The two definite rows of the table are the quaternion and anti-quaternion ones, and the four rows that carry an isotropic cone have indefinite automorphism groups.
Worked Examples
A null element of the realified centre. For $\tilde Q=e_t+ie_t$ the realified restriction of the centre gives $1^{2}-1^{2}=0$, while the complex form gives $\langle\tilde Q\rangle_{\natural}=(1+i)^{2}=2i\neq0$: the element is null for the realified form and is not a zero divisor, which is the pair of readings the centre section records.
A pure zero divisor. For $\tilde Q=\vec e_x+i\vec e_y$ the norm is $\langle\cdot\rangle_{\natural}=1+i^{2}=0$, and the element lies on the isotropic cone of the vector subspace, on the conic $\sum_kQ_k^{2}=0$ in $\mathbb{P}^{2}$; it is the line of the vector-subspace table.
An element of a definite row. For $\tilde Q=e_t+\vec e_x$ in the quaternion subspace the norm is $\langle\cdot\rangle_{\natural}=2$, and for $\tilde Q=ie_t+i\vec e_x$ in the anti-quaternion subspace it is $\langle\cdot\rangle_{\natural}=-2$: both are units, in agreement with the definiteness of the two rows, where no nonzero element is null.
The twist on a definite row. For $\tilde Q=e_t+\vec e_x$ the quaternion form gives $2$ and the $\varepsilon$-twisted form gives $1-1=0$, so the same element is a unit of the first form and a null element of the second: the exchange of the definite rows is visible on a single element, and it is the sign of the second coordinate that is responsible.
A non-pure zero divisor. For $\tilde Q=e_t+i\vec e_x$ the norm is $\langle\cdot\rangle_{\natural}=1+i^{2}=0$, and the element lies on the real null cone of the Hermitian subspace $\mathbb{M}_+$, of signature $(1,3)$; it is a zero divisor that is neither pure nor in the vector subspace, the two statements being the classification of Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors.
A maximal definite subspace of an indefinite row. In the vector subspace, of signature $(3,3)$, the span $\mathbb{R}\{\vec e_x,\vec e_y,\vec e_z\}$ is a maximal positive definite subspace of dimension $3$ and the span $\mathbb{R}\{i\vec e_x,i\vec e_y,i\vec e_z\}$ a maximal negative definite one; the two are orthogonal for the realified form, since the pairing of a real direction with its imaginary companion is purely imaginary, and are exchanged by multiplication by $i$, which is the local form of the exchange of the two definite rows of the whole algebra.
An automorphism of a restriction that does not extend. The real orthogonal group $O(4)$ of the quaternion subspace contains the reflection in the direction $\vec e_x$, but no element of the ambient automorphism group $O_4(\mathbb{C})$ restricts to it and fixes the anti-quaternion subspace pointwise, because the two definite rows are not orthogonal for the general quaternionic bilinear form (the pairing of $\vec e_x$ with $i\vec e_x$ is $i$); a form-preserving map that fixes one row pointwise is therefore constrained on the other, and the restriction groups of the table are larger than the restrictions of the ambient group.
The definite rows are not the real forms. The two definite rows are $\mathbb{H}_{\mathbb B}$, on which the form is $\sum_\mu q_\mu^{2}$, all signs $+1$, and $i\mathbb{H}_{\mathbb B}$, on which it is $-\sum_\mu q_\mu^{2}$, all signs $-1$; the Hermitian and anti-Hermitian subspaces, by contrast, are indefinite of signatures $(1,3)$ and $(3,1)$. The reason is that the Hermitian subspace $\mathbb{M}_+=\mathbb{R}\{e_t,i\vec e_x,i\vec e_y,i\vec e_z\}$ collects the single positive direction $e_t$ with the three negative directions $i\vec e_k$, while the anti-Hermitian subspace $\mathbb{M}_-=\mathbb{R}\{ie_t,\vec e_x,\vec e_y,\vec e_z\}$ collects the single negative direction $ie_t$ with the three positive directions $\vec e_k$; neither is spanned by directions of a single sign, which is what definiteness requires.
A null element of the anti-Hermitian subspace. For $\tilde Q=ie_t+\vec e_x$ the norm is $\langle\cdot\rangle_{\natural}=i^{2}+1=0$, on the real null cone of $\mathbb{M}_-$, of signature $(3,1)$; the sign of the null direction is the one the swapped table prescribes.
Summary
On the remarkable real subspaces the general quaternionic bilinear form carries the signatures $(1,1)$, $(3,3)$, $(4,0)$, $(0,4)$, $(1,3)$ and $(3,1)$, with the restriction Gram matrices $\operatorname{diag}(1,-1)$, $\operatorname{diag}(I_3,-I_3)$, $I_4$, $-I_4$, $\operatorname{diag}(1,-1,-1,-1)$ and $\operatorname{diag}(-1,1,1,1)$ in the natural real bases. The quaternion and anti-quaternion subspaces are the two definite rows, of dimension $4$ and maximal, and they are orthogonal complements for the realified form and exchanged by multiplication by $i$, which reverses the sign of the pairing; the centre and the vector subspace carry $(1,1)$ and $(3,3)$; the two real forms carry $(1,3)$ and $(3,1)$. The isotropic elements are the pure zero divisors on the vector subspace – the complex cone $\sum_kQ_k^2=0$, whose isotropic complex lines form a conic – and the non-pure zero divisors on the two real forms, whose isotropic real lines are parametrised by the real unit directions; the centre is indefinite but anisotropic over $\mathbb{C}$, and the two definite subspaces contain no isotropic element. The automorphism group of each restriction is the real orthogonal group of its signature, $O(1,1)$, $O(3,3)$, $O(4)$, $O(4)$, $O(1,3)$ and $O(3,1)$. The other three pairings on the same remarkable subspaces are Remarkable Subspaces and the Four General Products, Forms and Norms, Remarkable Subspaces under the General Plain Sesqualgebra and Remarkable Subspaces under the General Quaternionic Sesqualgebra, where the transposition of this table with the general plain bilinear one is read on the four real forms.
Summary of Notation
| Symbol | Meaning |
|---|---|
| $(1,1),(3,3),(4,0),(0,4),(1,3),(3,1)$ | the signatures of the norm |
| $\operatorname{diag}(I_3,-I_3)$, $I_4$, $-I_4$ | the restriction Gram matrices of the vector, quaternion and anti-quaternion rows |
| $\operatorname{diag}(I_4,-I_4)$ | the Gram matrix of the realified form in the grouped real basis $e_t,\dots,\vec e_z,ie_t,\dots,i\vec e_z$ |
| $\mathbb{R}(e_t\pm ie_t)$ | the two null lines of the realified centre |
| $\mathbb H_{\mathbb B}\oplus i\mathbb H_{\mathbb B}$ | the maximal definite splitting of the algebra |
| $\langle i\tilde Q_1,i\tilde Q_2\rangle_{\natural}=-\langle\tilde Q_1,\tilde Q_2\rangle_{\natural}$ | the sign change pairing the definite rows |
| $O(1,1),O(3,3),O(4),O(4),O(1,3),O(3,1)$ | the automorphism groups of the restrictions |
Further Reading
- Remarkable Subspaces and the Four General Products, Forms and Norms, for the companion table of the same remarkable subspaces
- Remarkable Subspaces under the General Plain Sesqualgebra, Remarkable Subspaces under the General Quaternionic Sesqualgebra, for the same remarkable subspaces under the other readings
- The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra, for the four forms on the remarkable subspaces, side by side
- Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, for the norm and its real forms
- The Isotropic Structure of the General Quaternionic Algebra, for the null cone on the subspaces
- Two-Sided Operators on the General Quaternionic Algebra, for the operators that preserve the form