The 2 Algebraic Norms over the Quaternioid $\mathbb{R}$ Space

Introduction

The two forms of The 2 Forms over the Quaternioid $\mathbb{R}$ Space are the scalar parts of the two general products, and their algebraic norms are the two algebraic norms of the chapter:

$$ \langle\tilde q\rangle=\langle\tilde q,\tilde q\rangle=q_t^2-\lvert\vec q\rvert^2 , \qquad \langle\tilde q\rangle_{\natural}=\langle\tilde q,\tilde q\rangle_{\natural}=q_t^2+\lvert\vec q\rvert^2 . $$

The plain algebraic norm is the plain form read at its two equal arguments, $\langle\tilde q\rangle=q_t^2-\lvert\vec q\rvert^2$, and the quaternion algebraic norm is the quaternion form read at its two equal arguments, $\langle\tilde q\rangle_{\natural}=q_t^2+\lvert\vec q\rvert^2$. The first is indefinite: it is $+1$ at $e_t$ and $-1$ at $\vec e_x$, so it is of both signs. The second is positive definite and vanishes only at the origin. Read on the products, the two are the coefficients of the two squares, $\tilde q\tilde q=\langle\tilde q\rangle e_t+2q_t\vec q$ and $\tilde q^{\natural}\tilde q=\langle\tilde q\rangle_{\natural}e_t$. The two norms are read off the two forms, the first from the plain form and the second from the quaternion form.

This article gives the two norms, the multiplicativity of the second and the failure of the first, the norm of a product against the product of the norms, the level sets read algebraically, and the invertibility criteria the two give. It assumes the two forms from The 2 Forms over the Quaternioid $\mathbb{R}$ Space and the two squares from Comparison Between the Two General Products, both preceding this one; the norms read on the two remarkable subspaces are Remarkable Subspaces and the Two Algebraic Norms, below this article in this category; and the theory of the norm and the invertibility of the plain product is owned by Norm and Invertibility in the Quaternion Algebra, in the Quaternions category.

The Two Norms

Definition. The two algebraic norms are the two forms read at their two equal arguments,

$$ \langle\tilde q\rangle=\langle\tilde q,\tilde q\rangle , \qquad \langle\tilde q\rangle_{\natural}=\langle\tilde q,\tilde q\rangle_{\natural} , $$

and on the coordinates

$$ \langle\tilde q\rangle=q_t^2-\lvert\vec q\rvert^2 , \qquad \langle\tilde q\rangle_{\natural}=q_t^2+\lvert\vec q\rvert^2. $$

The plain norm is indefinite, of both signs, and the quaternion norm is positive definite, zero only at the origin. Read on the products, the two squares are $\tilde q\tilde q=\langle\tilde q\rangle e_t+2q_t\vec q$, a general element of $\mathbb{H}$ whose scalar coefficient is the plain norm, and $\tilde q^{\natural}\tilde q=\langle\tilde q\rangle_{\natural}e_t$, a central element whose coefficient is the quaternion norm.

Multiplicativity

Theorem (multiplicativity of the quaternion norm; failure of the plain one). For all $\tilde q_1,\tilde q_2$,

$$ \boxed{\;\langle\tilde q_1\tilde q_2\rangle_{\natural}=\langle\tilde q_1\rangle_{\natural}\langle\tilde q_2\rangle_{\natural} , \qquad \text{equivalently}\qquad \bigl(\tilde q_1\tilde q_2\bigr)^{\natural}\bigl(\tilde q_1\tilde q_2\bigr)=\tilde q_1^{\natural}\tilde q_1\,\tilde q_2^{\natural}\tilde q_2\;}, $$

while the plain norm is not multiplicative:

$$ \boxed{\;\langle\tilde q_1\tilde q_2\rangle\neq \langle\tilde q_1\rangle\langle\tilde q_2\rangle \quad\text{in general} , \qquad \bigl(\vec e_x \vec e_y\bigr)^{2}=-e_t , \quad \vec e_x^2 \vec e_y^2=e_t\;}. $$

Proof. The quaternion norm is the scalar coefficient of the product of the element with its quaternion conjugate, $\langle\tilde q_1\rangle_{\natural}=\mathrm{Sc}\bigl(\tilde q_1^{\natural}\tilde q_1\bigr)$, and the conjugation is an anti-automorphism, so $\bigl(\tilde q_1\tilde q_2\bigr)^{\natural}\bigl(\tilde q_1\tilde q_2\bigr)=(\tilde q_1\tilde q_2)^{\natural}\tilde q_1\tilde q_2=\tilde q_2^{\natural}\tilde q_1^{\natural}\tilde q_1\tilde q_2=\tilde q_2^{\natural}|\tilde q_1|^2\tilde q_2=|\tilde q_1|^2|\tilde q_2|^2e_t$, the scalar $|\tilde q_1|^2$ commuting with every element; reading the scalar coefficient gives the multiplicativity. For the plain norm, the value of $\tilde q_1\tilde q_2\tilde q_1\tilde q_2$ is the square of the product and the two factors' squares do not determine it: at $\tilde q_1=\vec e_x,\tilde q_2=\vec e_y$ the product is $\vec e_z$, of square $-e_t$, while the product of the squares is $(-e_t)(-e_t)=e_t$, so the plain norm of the product differs from the product of the plain norms.

So the quaternion norm is multiplicative: the norm of a product is the product of the norms, and the quaternion norm is the reduced norm of the algebra, the square root of the determinant of the regular representation. The plain norm is not multiplicative, and this is the reason it is called a norm rather than the norm of the algebra: the plain product fails associativity in no way, but its algebraic norm is not a multiplicative function. The multiplicativity of the quaternion norm is the algebra's composition law for the four-square identity.

The Norm of a Product Against the Product of the Norms

Corollary (the two comparisons). For all $\tilde q_1,\tilde q_2$,

$$ \langle\tilde q_1\tilde q_2\rangle_{\natural}=\langle\tilde q_1\rangle_{\natural}\langle\tilde q_2\rangle_{\natural} , \qquad \text{equivalently}\qquad \bigl(\tilde q_1\tilde q_2\bigr)^{\natural}\bigl(\tilde q_1\tilde q_2\bigr)=\tilde q_1^{\natural}\tilde q_1\,\tilde q_2^{\natural}\tilde q_2 , $$

the quaternion norm of a product being exactly the product of the quaternion norms; and the plain norm of a product is not the product of the plain norms, the two squares differing by the commutator, $(\tilde q_1\tilde q_2)^2-\tilde q_1^2\tilde q_2^2=\tilde q_1[\tilde q_2,\tilde q_1]\tilde q_2$. The quaternion norm of the product is the same in the two orders, $\langle\tilde q_1\tilde q_2\rangle_{\natural}=\langle\tilde q_2\tilde q_1\rangle_{\natural}$, both being the value of the quaternion form on the product.

Proof. The first identity is the multiplicativity theorem. For the second, the plain square of the product is $(\tilde q_1\tilde q_2)^2$ and the product of the plain squares is $\tilde q_1^2\tilde q_2^2$, the two differing by $\tilde q_1[\tilde q_2,\tilde q_1]\tilde q_2$; at $\tilde q_1=\vec e_x,\tilde q_2=\vec e_y$ the two values are $-e_t$ and $+e_t$. For the third, $\langle\tilde q_1\tilde q_2\rangle_{\natural}=|\tilde q_1\tilde q_2|^2=|\tilde q_1|^2|\tilde q_2|^2=|\tilde q_2|^2|\tilde q_1|^2=\langle\tilde q_2\tilde q_1\rangle_{\natural}$, the coefficient being a real number.

So the quaternion norm is insensitive to the order of the factors and to the nesting of a product, while the plain norm is sensitive to both; the quaternion norm is the one that behaves as a norm of an algebra, and it is the one used in the invertibility criterion.

The Level Sets

Definition. A level set of a norm is the set of elements on which that norm takes a fixed value, $\langle\tilde q\rangle=c$ or $\langle\tilde q\rangle_{\natural}=c$ for a real $c$; the level sets are read algebraically, each being cut out by the equation on the coordinates.

Proposition (the level sets of the two norms).

The quaternion norm. For a real value $c$, $\langle\tilde q\rangle_{\natural}=c$ is the three-sphere $q_t^2+\lvert\vec q\rvert^2=c$ of radius $\sqrt c$ when $c>0$, the origin alone when $c=0$, and is empty when $c<0$.

The plain norm. For a real value $c$, $\langle\tilde q\rangle=c$ reads $q_t^2-\lvert\vec q\rvert^2=c$: a two-sheeted hyperboloid about the scalar axis when $c>0$, a one-sheeted hyperboloid when $c<0$, and the isotropic cone $q_t^2=\lvert\vec q\rvert^2$ when $c=0$.

Proof. The two equations are the two forms read at their two equal arguments. The quaternion equation is the sphere of radius $\sqrt c$ in the four coordinates, empty for $c<0$. The plain equation is the level set of the quadratic form $q_t^2-\lvert\vec q\rvert^2$, of signature $(1,3)$: a two-sheeted hyperboloid for a positive value, a one-sheeted hyperboloid for a negative one, and the cone for the vanishing value.

So the level sets of the two norms are of different shapes: the quaternion norm cuts the space into concentric three-spheres, cut by the origin and empty for the negative values; the plain norm cuts it into the hyperboloids of the form of signature $(1,3)$, with the isotropic cone as its zero level set. In particular the elements of unit quaternion norm form the unit three-sphere of the four coordinates, and the nonzero elements of zero plain norm are the generators of the cone, of the form $e_t\pm\hat u$ with $(\hat u,\hat u)=1$, which are the isotropic elements of the plain form.

The Invertibility Criteria

Theorem (the criterion). An element $\tilde q$ is invertible in the plain algebra if and only if $\langle\tilde q\rangle_{\natural}\neq0$, that is $\tilde q\neq0$; and then $\tilde q^{-1}=\tilde q^{\natural}/\lvert\tilde q\rvert^2$. The plain norm $\langle\tilde q\rangle$ gives no criterion of this kind: it is not multiplicative, and it vanishes at the nonzero invertible element $e_t+\vec e_x$, so the invertibility of a product cannot be tested on the plain norm.

Proof. The identity $\tilde q^{\natural}\tilde q=\langle\tilde q\rangle_{\natural}e_t$ shows that a nonzero $\tilde q$ has the inverse $\tilde q^{\natural}/\lvert\tilde q\rvert^2$, and a zero divisor would give $\langle\tilde q\rangle_{\natural}\langle\tilde q\rangle_{\natural}=0$ with both factors nonzero, impossible; so $\tilde q$ is invertible exactly when $\langle\tilde q\rangle_{\natural}\neq0$, that is exactly when $\tilde q\neq0$, and the criterion is the quaternion norm's. At $\tilde q=e_t+\vec e_x$ the plain norm is $\langle\tilde q\rangle=0$ while the quaternion norm is $\langle\tilde q\rangle_{\natural}=2$ and the element is invertible, so the vanishing of the plain norm decides nothing; the criterion is therefore the quaternion norm's alone.

So the quaternion norm is the norm of the algebra: it is real-valued, definite and multiplicative, and its vanishing gives the invertibility criterion and the division property; the plain norm is the algebraic norm of the plain product and is a norm only in the weak sense of the corpus's vocabulary.

Summary

The quaternion space carries two algebraic norms, the plain norm $\langle\tilde q\rangle=q_t^2-\lvert\vec q\rvert^2$ and the quaternion norm $\langle\tilde q\rangle_{\natural}=q_t^2+\lvert\vec q\rvert^2$, the two forms read at their two equal arguments, the first from the plain form and the second from the quaternion form. The first is indefinite, taking $+1$ and $-1$ at $e_t$ and $\vec e_x$; the second is positive definite, zero only at the origin. The second is multiplicative, $\langle\tilde q_1\tilde q_2\rangle_{\natural}=\langle\tilde q_1\rangle_{\natural}\langle\tilde q_2\rangle_{\natural}$, and the first is not, $\langle\tilde q_1\tilde q_2\rangle\neq\langle\tilde q_1\rangle\langle\tilde q_2\rangle$ in general; read on the squares, the witness is $\bigl(\vec e_x \vec e_y\bigr)^{2}=-e_t$ against $\vec e_x^2 \vec e_y^2=e_t$. The quaternion norm of a product equals the product of the quaternion norms and does not depend on the order, while the plain norm of a product is not the product of the plain norms.

The level sets of the quaternion norm are the three-spheres, cut by the origin and empty for the negative values; those of the plain norm are the hyperboloids of the form of signature $(1,3)$, with the isotropic cone as the zero level set. The invertibility criterion is the quaternion norm's: $\tilde q$ is invertible exactly when $\langle\tilde q\rangle_{\natural}\neq0$, with inverse $\tilde q^{\natural}/\lvert\tilde q\rvert^2$; the plain norm gives no criterion, vanishing at the nonzero element $e_t+\vec e_x$.

Summary of Notation

Symbol	Meaning
$\langle\tilde q\rangle=\langle\tilde q,\tilde q\rangle=q_t^2-\lvert\vec q\rvert^2$	the plain algebraic norm, indefinite
$\langle\tilde q\rangle_{\natural}=\langle\tilde q,\tilde q\rangle_{\natural}=q_t^2+\lvert\vec q\rvert^2$	the quaternion algebraic norm, positive definite
$\tilde q^{\natural}\tilde q=\langle\tilde q\rangle_{\natural}e_t$	the quaternion norm read on the product
$\langle\tilde q_1\tilde q_2\rangle_{\natural}=\langle\tilde q_1\rangle_{\natural}\langle\tilde q_2\rangle_{\natural}$	the multiplicativity of the quaternion norm
$\bigl(\vec e_x \vec e_y\bigr)^{2}=-e_t$, $\vec e_x^2 \vec e_y^2=e_t$	the failure of the plain norm
$q_t^2+\lvert\vec q\rvert^2=c$: the three-sphere	the level sets of the quaternion norm
$q_t^2-\lvert\vec q\rvert^2=c$: the hyperboloids	the level sets of the plain norm
$\tilde q^{-1}=\tilde q^{\natural}/\lvert\tilde q\rvert^2$, $\tilde q\neq0$	the invertibility criterion