Square Roots of a General Element in the General Quaternionic Algebra
Introduction
This article solves the square-root equation $\tilde\Xi\star\tilde\Xi=\tilde Q$ in the general quaternionic bilinear product for an arbitrary element $\tilde Q$ of the biquaternion algebra, and it states the absence that the equation forces.
The equation names a product, and throughout this article the square is the one of the general quaternionic product; the plain square of the same element is a different equation, and its solution is Square Roots of a General Element in the Biquaternion Algebra. The quaternionic square is the central element $\langle\tilde\Xi\rangle_{\natural}e_t$ for every $\tilde\Xi$, so the equation can be solved only when the right-hand side is a central value, and this is the first statement of the article: an element which is not a central value has no square root in this product, and the absence is stated and not filled. For a central radicand $\tilde Q=\lambda e_t$ the equation reduces to the single complex equation $\langle\tilde\Xi\rangle_{\natural}=\lambda$, and the root set is a nondegenerate complex quadric, of complex dimension three when $\lambda\neq0$ and the zero-divisor cone when $\lambda=0$. Every central value has roots, and the root set of $\lambda$ is a scaled copy of the root set of $1$ or of $-1$.
The results. The quaternionic square is always central, $\tilde\Xi\star\tilde\Xi=\langle\tilde\Xi\rangle_{\natural}e_t$, so the equation $\tilde\Xi\star\tilde\Xi=\tilde Q$ has a solution if and only if $\tilde Q$ is a central value: an element with nonzero vector part has no square root in this product. For a central radicand $\tilde Q=\lambda e_t$ the root set is $R(\lambda)=\{\tilde\Xi:\langle\tilde\Xi\rangle_{\natural}=\lambda\}$, which is a nondegenerate complex quadric of complex dimension $3$ and real dimension $6$ when $\lambda\neq0$, and the zero-divisor cone of real dimension $6$ when $\lambda=0$, and it is never empty. The sets are not new: $R(\lambda)=\alpha R(1)=(i\alpha)R(-1)$ for $\alpha^2=\lambda\neq0$, so the three central values generate every value, and a root of a nonzero value is a unit of the algebra while a nonzero root of $0$ is a zero divisor. Each section below states its result first and then proves it.
The Reduction
Definition. Let $\tilde Q\in\mathbb{B}$. A square root of $\tilde Q$ in the general quaternionic product is an element $\tilde\Xi$ with
$$ \tilde\Xi\star\tilde\Xi=\tilde Q . $$
The root is relative to the product, and this article uses the one of $\star$ throughout; the general definition of the square and of the roots of an element is Definitions for the Study of the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, §Powers, the Square and the Roots.
Theorem (the square is central). For every $\tilde\Xi\in\mathbb{B}$ the square $\tilde\Xi\star\tilde\Xi$ lies in the scalar line $\mathbb{C}_{\mathbb{B}}$, and
$$ \boxed{\; \tilde\Xi\star\tilde\Xi=\langle\tilde\Xi\rangle_{\natural}e_t \;}. $$
Proof. The scalar–vector form of the product gives $\tilde\Xi^{\natural}\tilde\Xi=(\Xi_t-\vec\Xi)(\Xi_t+\vec\Xi)=\Xi_t^2-\vec\Xi^2=\bigl(\Xi_t^2+(\vec\Xi,\vec\Xi)\bigr)e_t$; the statement is §The Square Lies in the Scalar Line of Introduction to the General Quaternionic Algebra.
Theorem (the existence criterion). The equation $\tilde\Xi\star\tilde\Xi=\tilde Q$ has a solution if and only if $\tilde Q$ is a central value. For $\tilde Q=\lambda e_t$ the equation is equivalent to the single complex equation
$$ \langle\tilde\Xi\rangle_{\natural}=\lambda . $$
Proof. A square is central by the preceding theorem, so a solution forces $\tilde Q\in\mathbb{C}_{\mathbb{B}}$; conversely if $\tilde Q=\lambda e_t$ the equation reads $\langle\tilde\Xi\rangle_{\natural}e_t=\lambda e_t$, which is $\langle\tilde\Xi\rangle_{\natural}=\lambda$ because $e_t\neq0$.
Absence (the non-central radicands). An element of $\mathbb{B}$ whose vector part is nonzero has no square root in the general quaternionic product. The witness $\vec e_x$ shows it: a root would satisfy $\langle\tilde\Xi\rangle_{\natural}e_t=\vec e_x$, and the two sides differ in the coefficient of $\vec e_x$. This is not a deficiency of a particular element but of the product, whose squaring map has image the scalar line; the plain product has a wider image, and its squaring map reaches every element, which is why the plain problem has a solution for an arbitrary radicand.
The Roots of a Central Value
For a central radicand $\tilde Q=\lambda e_t$ the root set is $R(\lambda)=\{\tilde\Xi:\langle\tilde\Xi\rangle_{\natural}=\lambda\}$: a nondegenerate complex quadric of real dimension six when $\lambda\neq0$, the zero-divisor cone of real dimension six when $\lambda=0$, and never empty.
Theorem (the root set of a central value). Let $\lambda\in\mathbb{C}$ and let $\tilde Q=\lambda e_t$. The root set of $\tilde Q$ is
$$ \boxed{\; R(\lambda)=\{\tilde\Xi\in\mathbb{B}:\langle\tilde\Xi\rangle_{\natural}=\lambda\} \;}. $$
- If $\lambda\neq0$, $R(\lambda)$ is a nondegenerate complex quadric of complex dimension three and real dimension six.
- If $\lambda=0$, $R(0)$ is the zero-divisor cone, a real cone of real dimension six.
- The set is never empty: the element $\tilde\Xi=\sqrt\lambda\,e_t$ is a root, where $\sqrt\lambda$ is either square root of $\lambda$ in $\mathbb{C}$.
Proof. The identification with the level set is the reduction. With $\Xi_\mu=p_\mu+ip'_\mu$, the equation $N=\sum_\mu(p_\mu^2-(p'_\mu)^2)+2i\sum_\mu p_\mu p'_\mu=\lambda$ is a pair of real equations, the gradient of $N$ does not vanish off the origin, and the level set has real dimension $8-2=6$ in the two cases; for $\lambda=0$ it is a cone because $\langle\cdot\rangle_{\natural}$ is homogeneous. The displayed central element has $\langle \sqrt\lambda\,e_t\rangle_{\natural}=(\sqrt\lambda)^2=\lambda$.
Corollary (the central roots). The root set of every central value contains the two central roots $\pm\sqrt\lambda\,e_t$, and they coincide exactly when $\lambda=0$; for $\lambda\neq0$ the two are distinct and lie in the scalar line $\mathbb{C}_{\mathbb{B}}$.
The Scaling of the Root Set
For $\lambda\neq0$ and $\alpha^2=\lambda$ the root set of $\lambda$ is the scaled root set of $1$ and equally of $-1$, $R(\lambda)=\alpha R(1)=(i\alpha)R(-1)$.
Theorem. Let $\lambda\neq0$ and let $\alpha\in\mathbb{C}$ satisfy $\alpha^2=\lambda$. Then multiplication by $\alpha$ is a bijection of the roots of $1$ onto the roots of $\lambda$, and multiplication by $i\alpha$ is a bijection of the roots of $-1$ onto the roots of $\lambda$:
$$ \boxed{\; R(\lambda)=\alpha\,R(1)=(i\alpha)\,R(-1) \;}. $$
Proof. The algebraic norm is homogeneous of degree two and $\mathbb{C}$-linear in the scaling, $\langle\alpha\tilde E\rangle_{\natural}=\alpha^2\langle\tilde E\rangle_{\natural}$, so $\alpha$ carries the level set $N=1$ onto the level set $N=\alpha^2=\lambda$; the scaling is a bijection of $\mathbb{B}$ because $\alpha\neq0$. For the second display, $(i\alpha)^2=-\alpha^2=-\lambda$, so $i\alpha$ carries the level set $N=-1$ onto the level set $N=(-1)(-\lambda)=\lambda$.
The root set of a general central value is therefore not a new object: it is the root set of $1$, or equivalently that of $-1$, scaled by a square root of the value. The three special values of Square Roots of One and the Idempotents of the General Quaternionic Algebra and Square Roots of Zero and the Nilpotents of the General Quaternionic Algebra generate every other central value, and the cone is the remaining case.
The Classification
Every element falls into exactly one row: a non-central radicand has an empty root set, a nonzero central value has the quadric $N=\lambda$ of real dimension six, and the central value $0$ has the zero-divisor cone of real dimension six.
| radicand $\tilde Q$ | condition | root set |
|---|---|---|
| $\tilde Q\notin\mathbb{C}_{\mathbb{B}}$ | $\tilde Q$ not central | empty; the absence of §The Reduction |
| $\tilde Q=\lambda e_t$, $\lambda\neq0$ | a non-zero central value | the nondegenerate quadric $N=\lambda$, real dimension six, the scaled root set of $1$ |
| $\tilde Q=0$ | the central value $0$ | the zero-divisor cone $N=0$, real dimension six |
The classification covers every element; the two non-empty rows differ in whether the level set is a nondegenerate quadric or a cone, and they agree in real dimension. The plain product's classification is a different shape: there the root set is the central pair $\pm\sqrt\lambda\,e_t$ together with the quadric cut on the vector subspace, the two pieces distinguished by the branch of the vector–scalar reduction (Square Roots of a General Element in the Biquaternion Algebra).
The Three Central Values
The values $-1$, $0$ and $+1$ are the seeds of the classification: $R(-1)$ is the quadric $N=-1$, $R(+1)$ the quadric $N=1$, and $R(0)$ the zero-divisor cone, and every other central value is a scaling of the first or the third.
The values $\lambda=-1$, $0$ and $+1$ are the subject of the companion articles Square Roots of One and the Idempotents of the General Quaternionic Algebra and Square Roots of Zero and the Nilpotents of the General Quaternionic Algebra, and they are the seeds of the classification:
- $R(-1)=\{N=-1\}$, a quadric of real dimension six containing the central roots $\pm i$ and the imaginary vector roots $\tilde\Xi$ over the real unit vectors $\vec u$;
- $R(0)=\{N=0\}=C$, the zero-divisor cone, containing the origin, the pure isotropic elements and the complex multiples of the nontrivial plain idempotents;
- $R(+1)=\{N=1\}$, a quadric of real dimension six containing the central roots $\pm e_t$ and the real vector roots $\vec u$.
By the scaling theorem, $R(\lambda)$ for a general value is the image of one of the first and the third under multiplication by a square root of $\lambda$, and the second is the cone, which is fixed by every non-zero scaling.
Worked Examples
- Roots of $1$. The central roots are $\pm e_t$. The element $\vec e_x$ is a root, since $\langle \vec e_x\rangle_{\natural}=1$; more generally every real unit vector $\vec u$ is a root, and the elements $i\vec q$ with $(\vec q,\vec q)=-1$ are roots as well. The element $e_t+\vec e_x$ is not a root, since $\langle e_t+\vec e_x\rangle_{\natural}=1+1=2$.
- Roots of $-1$. The central roots are $\pm i$, and the elements $\tilde\Xi$ over the real unit vectors are roots. The element $\vec e_x$ is not a root of $-1$; it is a root of $+1$. This is the sign that the quaternionic product removes from the pure square.
- Roots of $0$. The origin, the pure isotropic element $\vec e_x+i\vec e_y$, and the non-pure element $e_t+i\vec e_x$ are roots; the first is the trivial one, the second is a nilpotent of the plain product, and the third is twice a Hermitian idempotent. The root set is the whole cone, and its members are the zero divisors of the algebra.
- Roots of a complex value. For $\lambda=4$ the element $2e_t$ is a root, and so is $2\vec e_x$; for $\lambda=i$ the element $\alpha e_t$ with $\alpha^2=i$, for instance $\alpha=(1+i)/\sqrt2$, is a root, and the whole root set is the image of $R(1)$ under multiplication by $\alpha$.
- A non-central radicand. The element $\vec e_x$ has no root: its square-root equation would force the central element $\langle\tilde\Xi\rangle_{\natural}e_t$ to equal $\vec e_x$. The statement is the absence of §The Reduction, and it is the sharpest difference from the plain product, where $\vec e_x$ has the four roots $\pm\tfrac{1}{\sqrt2}(e_t+\vec e_x)$ and $\pm\tfrac{i}{\sqrt2}(e_t-\vec e_x)$ (Square Roots of a General Element in the Biquaternion Algebra).
The Roots Which Are Units, and the Roots Which Are Zero Divisors
A root of a nonzero central value has $N\neq0$ and is a unit of the plain product satisfying the two one-sided equations of $\star$, while a nonzero root of $0$ has $N=0$ and is a zero divisor with neither equation soluble.
The algebraic norm decides the nature of every root, and it decides it the same way as in the other products. A root of a non-zero central value has $\langle\tilde\Xi\rangle_{\natural}=\lambda\neq0$, so it is a unit of the plain product and satisfies the two one-sided equations of $\star$; a non-zero root of $0$ has $\langle\tilde\Xi\rangle_{\natural}=0$, so it is a zero divisor of the algebra, and neither one-sided equation is soluble for it (Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra). The statement is the block reading of the general theorem of Square Roots of a General Element in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, §The Roots Are Units, or Zero Divisors. In particular the roots of $-1$ and of $+1$ are never zero divisors, and the roots of $0$ other than the origin are always zero divisors; the origin is neither.
Summary
The square-root equation $\tilde\Xi\star\tilde\Xi=\tilde Q$ of the general quaternionic product has a solution exactly when $\tilde Q$ is a central value; an element with non-zero vector part has no root, because the quaternionic square is always central. For a central radicand the equation reduces to the single complex equation $\langle\tilde\Xi\rangle_{\natural}=\lambda$, the square of the equation being the algebraic norm, and the root set is the nondegenerate complex quadric $N=\lambda$ of real dimension six when $\lambda\neq0$, and the zero-divisor cone of real dimension six when $\lambda=0$. Every central value has roots, among them the central root $\sqrt\lambda\,e_t$, and the root set of $\lambda$ is the root set of $1$, or of $-1$, scaled by a square root of the value. The three special values generate the classification: $R(-1)$ contains the imaginary vector roots $\tilde\Xi$, $R(+1)$ the real vector roots $\vec u$, and $R(0)$ the whole nilpotent cone of the product. The roots of a non-zero value are units of the algebra, the non-zero roots of $0$ are its zero divisors, and the absence of a root for a non-central radicand is stated and not filled.
Summary of Notation
| symbol | meaning |
|---|---|
| $\tilde\Xi\star\tilde Q=\tilde\Xi^{\natural}\tilde Q$ | the general quaternionic bilinear product |
| $\langle\tilde\Xi\rangle_{\natural}=\sum_\mu \Xi_\mu^2=\Xi_t^2+(\vec\Xi,\vec\Xi)$ | the algebraic norm |
| $\mathbb{C}_{\mathbb{B}}=\mathbb{C}e_t$ | the scalar line, the set of the central values |
| $\lambda e_t$ | a central value, $\lambda=\lambda_1+i\lambda_2$ |
| $\tilde\Xi\star\tilde\Xi=\langle\tilde\Xi\rangle_{\natural}e_t$ | the central square |
| $R(\lambda)=\{N=\lambda\}$ | the root set of the central value $\lambda$ |
| $C=R(0)$ | the zero-divisor cone |
| $\alpha$, $\sqrt\lambda$ | the square roots of the value, with $\alpha^2=\lambda$ |
| $R(\lambda)=\alpha R(1)=(i\alpha)R(-1)$ | the scaling of the root set |
Further Reading
- Nathan Jacobson, Structure and Representations of Jordan Algebras (American Mathematical Society, 1968), for the square roots of an element in an algebra with a multiplicative quadratic form.
- Richard D. Schafer, An Introduction to Nonassociative Algebras (Academic Press, 1966), for the squaring map of a non-associative algebra and its image.
- Square Roots of a General Element in the Biquaternion Algebra, for the plain problem, its vector–scalar reduction and its two branches.
- Square Roots of One and the Idempotents of the General Quaternionic Algebra, for the roots of $+1$ and of $-1$, the idempotents and the displaced plain idempotents.
- Square Roots of Zero and the Nilpotents of the General Quaternionic Algebra, for the roots of $0$ and the nilpotent cone.
- Square Roots of a General Element in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the four root problems side by side.
- Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, for the algebraic norm, the one-sided equations and the set $\mathbb{B}^{\times}$.
- Zero Divisors of the General Quaternionic Algebra, for the cone and its two families.