Remarkable Subspaces under the General Quaternionic Sesqualgebra

Introduction

The general quaternionic sesquilinear form $\langle\tilde Q,\tilde Q\rangle_{\natural*}=\mathrm{Sc}(\tilde Q^{\natural}\tilde Q^{*})=\sum_\mu\varepsilon_\mu|Q_\mu|^{2}$ of The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra, of signature $(2,6)$, is read here on the remarkable real subspaces of Introduction to the Remarkable Subspaces — the centre $\mathbb{C}_{\mathbb{B}}$, the vector subspace $\mathrm{Vect}(\mathbb{B})$, the quaternion subspace $\mathbb{H}_{\mathbb{B}}$, the anti-quaternion subspace $i\mathbb{H}_{\mathbb{B}}$, the Hermitian subspace $\mathbb{M}_+$ and the anti-Hermitian subspace $\mathbb{M}_-$.

The form is used, not re-derived. Its definition, its Krein signature $(2,6)$ and its automorphism group $\mathrm{U}(1,3)$ are The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra; its null set and its totally isotropic subspaces are Krein Norm Regularity and Degeneracy in the Biquaternion Algebra; the remarkable subspaces and their bases are Introduction to the Remarkable Subspaces; and the companion readings of the same six by the other three pairings are Remarkable Subspaces and the Four General Products, Forms and Norms, Remarkable Subspaces under the General Quaternionic Algebra and Remarkable Subspaces under the General Plain Sesqualgebra. This article owns the restrictions themselves and the structure of each one.

The Restriction Matrices

The form differs from the Euclidean square by the sign vector $\varepsilon=(1,-1,-1,-1)$ on the vector directions, so on the remarkable subspaces it is either the Euclidean square, its negative, or a Lorentzian form. In the natural real basis of each subspace it is diagonal, and the diagonals are the sign strings below.

Proposition (the matrices). In the natural real basis displayed the restriction is diagonal with entries $\varepsilon_\mu$ on the four-dimensional rows and with entries $1,1$ on the centre and $-1$ on the vector subspace; each restriction is non-degenerate, with Gram determinant $\pm1$.

Proof. In the real basis of the algebra the form is diagonal with the signs $\varepsilon,\varepsilon$, since $\langle e_\mu,e_\nu\rangle_{\natural*}=\varepsilon_\mu\delta_{\mu\nu}$ and $\langle ie_\mu,ie_\nu\rangle_{\natural*}=\varepsilon_\mu|i|^{2}\delta_{\mu\nu}=\varepsilon_\mu\delta_{\mu\nu}$, while the mixed entries have vanishing real part. The natural real basis of each subspace is a subset of the algebra's real basis, so each restriction is the corresponding principal submatrix. On the centre the basis is $\{e_t,ie_t\}$, giving $\operatorname{diag}(1,1)$; on the vector subspace it is $\{\vec e_k,i\vec e_k\}$, where the signs are all $-1$; and on each four-dimensional subspace the basis selects one direction of each complex coordinate, giving the sign string $\varepsilon$ in a different order.

Remark (the four real forms carry the Lorentzian form). On each of the four four-dimensional real subspaces the restriction is a Lorentzian form of signature $(1,3)$: it is the Krein form of Lorentzian space, with one timelike and three spacelike directions. The four selections of the timelike direction are the four real subspaces: the real scalar direction $e_t$ on $\mathbb{H}_{\mathbb{B}}$, the imaginary scalar direction $ie_t$ on $i\mathbb{H}_{\mathbb{B}}$, the real scalar direction again on $\mathbb{M}_+$, and the imaginary scalar direction again on $\mathbb{M}_-$.

The Four Real Forms

Definition. A real form of the form is a real four-dimensional subspace on which the restriction is real-valued and symmetric. The four carried by the remarkable subspaces are the quaternion form $\mathbb{H}_{\mathbb{B}}$, the anti-quaternion form $i\mathbb{H}_{\mathbb{B}}$, the Hermitian form $\mathbb{M}_{+}$ and the anti-Hermitian form $\mathbb{M}_{-}$.

Theorem (the four real forms). On each of the four subspaces the form restricts to a real symmetric bilinear form of signature $(1,3)$, the Krein form of a Lorentzian four-space; multiplication by $i$ exchanges the two pairs, $\mathbb{H}_{\mathbb{B}}\leftrightarrow i\mathbb{H}_{\mathbb{B}}$ and $\mathbb{M}_{+}\leftrightarrow\mathbb{M}_{-}$. Each contains isotropic elements, and inside a form of signature $(1,3)$ the maximal totally isotropic real dimension is $\min(1,3)=1$.

Proof. The four rows of the restriction table are the four $(1,3)$ rows, and on each of them every coefficient pair $Q_{2\mu}^{*}Q_{1\mu}$ is real, because the coefficients are either both real or both purely imaginary; hence the form is real and symmetric. Multiplication by $i$ multiplies the scalar coefficient of an element of $\mathbb{M}_{+}$ by $i$ and leaves the vector coefficients, exchanging the two sectors, and carries the quaternion subspace onto the anti-quaternion subspace. The index bound is $\min(n,m)$ for the signature $(1,3)$.

Remark (the four are not a classification). The four subspaces are the real slices carried by the remarkable subspaces, and the two definite complex rows are not real forms: the centre has signature $(2,0)$ and the vector subspace $(0,6)$. Other real four-planes on which the form is real exist — for instance $\mathrm{span}_{\mathbb{R}}\{e_t,\vec e_x,i\vec e_y,i\vec e_z\}$ — so the four are a natural list, not an exhaustive one. The real basis of the algebra, in which the mixed entries are purely imaginary and the realified Gram matrix is $\operatorname{diag}(1,-1,-1,-1,1,-1,-1,-1)$, is The Realification of the General Quaternionic Sesqualgebra.

The Definite Rows and the Fundamental Decomposition

Two of the rows are definite. On the centre the form is positive definite, of signature $(2,0)$, and on the vector subspace it is negative definite, of signature $(0,6)$. The two are the fundamental decomposition of the Krein space,

$$ \boxed{\; \mathbb{B}=\mathbb{C}_{\mathbb{B}}\perp^{+}\mathrm{Vect}(\mathbb{B}), \qquad (2,0)+(0,6)=(2,6) \;}, $$

and they realise the whole positive index $2$ and the whole negative index $6$ of the ambient signature.

Proposition (the maximal definite dimensions). The maximal dimension of a positive definite subspace of the form is $2$, attained by the centre, and the maximal dimension of a negative definite subspace is $6$, attained by the vector subspace; the four remaining subspaces are indefinite of signature $(1,3)$.

Proof. The positive index of the form is $2$ and its negative index $6$, so a definite subspace has dimension at most the corresponding index; the centre and the vector subspace attain the bounds, being definite of those dimensions by the table. The four remaining rows are indefinite of signature $(1,3)$, so they contain directions of both signs.

Each of the four indefinite rows carries a real null cone, of real dimension $3$, which is the cone of the nonzero null elements of the row and the cone of the zero divisors that lie in it:

Remark (the isotropic lines of a row). On each indefinite row the null cone is a quadratic cone of real dimension $3$, whose isotropic real lines are parametrised by the real unit directions; the maximal dimension of a totally isotropic subspace is $1$ on each row, since the signature is $(1,3)$ and the Witt index of a Lorentzian form is $1$. On the two definite rows there is no isotropic element, and on the centre and the vector subspace the form is definite, so the origin is the only isotropic point. The isotropic structure of the form on the whole algebra, where the Witt index over $\mathbb{R}$ is $2$, is The Krein Norm-Degenerate Sector, §The Witt Index.

The Coincidences with the Other Three Forms

On the remarkable subspaces the general quaternionic sesquilinear form coincides with one of the other pairings, up to a sign, on every row:

Proof. On the centre the sign $\varepsilon_t=1$ leaves $\lvert Q_t\rvert^{2}$, which is the general plain sesquilinear restriction; on the vector subspace the three signs $\varepsilon_k=-1$ give $-\sum_k\lvert Q_k\rvert^{2}$, the negative of the general plain sesquilinear restriction. On $\mathbb{H}_{\mathbb{B}}$ the coefficients are real, so $\lvert Q_\mu\rvert^{2}=Q_\mu^{2}$ and the form reads $\sum_\mu\varepsilon_\mu Q_\mu^{2}$, which is the general plain bilinear restriction; on $i\mathbb{H}_{\mathbb{B}}$ the coefficients are $iq'_\mu$, so $\lvert Q_\mu\rvert^{2}=(q'_\mu)^{2}$ and the form reads $\sum_\mu\varepsilon_\mu(q'_\mu)^{2}$, again the general plain bilinear restriction, the two signs of $i^{2}$ cancelling. On $\mathbb{M}_+$ the scalar coefficient is real and the vector coefficients are $ip_k$, so the form reads $q_t^{2}-\sum_kp_k^{2}$, which is the general quaternionic bilinear restriction $\sum_\mu Q_\mu^{2}=q_t^{2}+\sum_k(ip_k)^{2}$; on $\mathbb{M}_-$ the scalar coefficient is $ib_t$ and the vector coefficients are real, so the form reads $b_t^{2}-\sum_kq_k^{2}$ while the general quaternionic bilinear restriction reads $-b_t^{2}+\sum_kq_k^{2}$, the negative.

Remark (the coincidence is not an identity). The coincidences are accidental to the subspace and do not extend to the algebra: the four forms are pairwise distinct there, of signatures $(4,4)$, $(4,4)$, $(8,0)$ and $(2,6)$. The comparison of the four on the same remarkable subspaces is The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra.

The Automorphism Groups of the Restrictions

Definition. For each subspace $V$ of the remarkable subspaces, the automorphism group of the restriction is the group of real-linear maps of $V$ preserving the restriction. Since each restriction is non-degenerate, it is the real orthogonal group of its signature.

Proof. A real-linear automorphism of a non-degenerate real symmetric form of signature $(n,m)$ is an element of $O(n,m)$, of real dimension $n(n-1)/2$ with $n=p+q$, giving $1$, $15$ and $6$; the two definite rows give $O(2)$ and $O(6)\cong O(0,6)$, and the four Lorentzian rows give $O(1,3)$ four times.

Remark (the ambient Krein group). The complex-linear automorphism group of the form on the algebra is the indefinite unitary group $\mathrm{U}(1,3)$ of real dimension $16$, of The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra; the real-linear automorphism group of the realified form is the larger $O(2,6)$. The groups of the table are the largest groups preserving the restriction of the form to a subspace, and they are not induced by the ambient group: an automorphism of the ambient form need not preserve a given subspace.

Worked Examples

A positive central element. For $\tilde Q=e_t$ the value is $\varepsilon_t|1|^{2}=1$, and for $\tilde Q=e_t+ie_t$ it is $1+1=2$, in agreement with the positive definiteness of the centre and its signature $(2,0)$.

A negative vector element. For $\tilde Q=\vec e_x$ the value is $\varepsilon_x|1|^{2}=-1$, and for $\tilde Q=\vec e_x+i\vec e_x$ it is $-1-1=-2$; the vector subspace is negative definite, and it realises the negative index $6$ of the form.

The null element of the quaternion row. For $\tilde Q=e_t+\vec e_x$ the value is $1-1=0$, and the element lies on the null cone $q_t^{2}=q_x^{2}+q_y^{2}+q_z^{2}$ of the quaternion subspace; it is a zero divisor of the algebra, and the Peirce lines it carries are those of The Krein Norm-Degenerate Sector, §The Doubly Null Stratum.

The null element of the anti-quaternion row. For $\tilde Q=ie_t+i\vec e_x$ the value is $1-1=0$, on the cone $(q'_t)^{2}=\sum_k(q'_k)^{2}$ of the anti-quaternion subspace; the sign of the scalar coefficient is the one the row prescribes.

The null element of the Hermitian row. For $\tilde Q=e_t+i\vec e_x$ the value is $1-1=0$, on the cone $q_t^{2}=\sum_k(q'_k)^{2}$ of $\mathbb{M}_+$; the same element is null for the general quaternionic bilinear form, with which the form coincides on this row.

The null element of the anti-Hermitian row. For $\tilde Q=ie_t+\vec e_x$ the value is $1-1=0$, on the cone $(q'_t)^{2}=\sum_kq_k^{2}$ of $\mathbb{M}_-$.

A maximal totally isotropic real line. On the quaternion subspace the real line $\mathbb{R}\{e_t+\vec e_x\}$ is a maximal totally isotropic real subspace, of the maximal dimension $1$ that the Witt index of the restriction allows; in the ambient complex space the complex line $\mathbb{C}(e_t+\vec e_x)$ is totally isotropic, and it meets the quaternion subspace in that real line.

The centre is not isotropic. For a central element $Ae_t$ with $A=q_t+iq'_t$ the value is $\lvert A\rvert^{2}=q_t^{2}+(q'_t)^{2}$, which vanishes only at $A=0$: the centre is the positive definite half of the fundamental decomposition, and its only isotropic point is the origin.

Summary

On the remarkable real subspaces the general quaternionic sesquilinear form carries the signatures $(2,0)$, $(0,6)$ and four copies of $(1,3)$, with the restriction matrices $\operatorname{diag}(1,1)$ on the centre, $-\mathrm{I}_6$ on the vector subspace and $\mathrm{E}=\operatorname{diag}(1,-1,-1,-1)$ on each of the four four-dimensional subspaces. The centre and the vector subspace are the definite rows, of dimensions $2$ and $6$ and opposite signs, and they are the fundamental decomposition of the Krein space, realising the positive index $2$ and the negative index $6$ of the ambient signature. The four remaining rows are Lorentzian, each with a real null cone of dimension $3$ whose isotropic lines are parametrised by the real unit directions; the maximal totally isotropic dimension is $1$ on each of them, and the form is definite on the two remaining rows. On all six the form coincides with one of the other pairings, up to a sign: with the general plain sesquilinear form on the centre, with the negative of it on the vector subspace, with the general plain bilinear form on the quaternion and anti-quaternion subspaces, and with the general quaternionic bilinear form, up to a sign, on the two Hermitian subspaces. The automorphism groups are $O(2)$, $O(6)$ and four copies of $O(1,3)$. The comparison with the three other pairings on the same remarkable subspaces is The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra.

Summary of Notation

Further Reading

• The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra, for the form, its coefficient Gram matrix and its inertia.
• The Krein Norm-Degenerate Sector, for the null set, the doubly null lines and the Witt index.
• Introduction to the Remarkable Subspaces, for the remarkable real subspaces and their bases
• Remarkable Subspaces under the General Plain Sesqualgebra, for the definite companion reading of the same six
• Remarkable Subspaces under the General Quaternionic Algebra, for the norm reading of the same six
• The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra, for the four forms on the remarkable subspaces side by side