Conditions Of "Norm" Isomorphism In Quaternion Algebra

Apr 22, 2025 by ADMIN 55 views

**Conditions of "Norm" isomorphism in Quaternion Algebra**

Introduction

In the realm of quaternion algebra, the concept of "norm" isomorphism plays a crucial role in understanding the properties and behavior of quaternions. The norm of a quaternion is a scalar value that represents the magnitude or size of the quaternion. In this article, we will delve into the conditions of "norm" isomorphism in quaternion algebra, exploring the underlying mathematical framework and its implications.

Quaternion Algebra

Quaternion algebra is a mathematical structure that extends the real numbers to include three additional imaginary units, denoted as i, j, and k. These imaginary units satisfy certain rules and relationships, which form the foundation of quaternion algebra. The quaternions are represented as:

q = w + xi + yj + zk

where w, x, y, and z are real numbers, and i, j, and k are the imaginary units.

Norm of a Quaternion

The norm of a quaternion q is defined as:

\widetilde{\mathrm{N}} q = \mathrm{T}\left(q^{2}\right)/2

where T denotes the trace of the quaternion. The norm of a quaternion represents its magnitude or size, and it plays a crucial role in many applications of quaternion algebra.

Conditions of "Norm" Isomorphism

The problem of determining the conditions of "norm" isomorphism in quaternion algebra is a fundamental question in the field. In this context, an isomorphism between two quaternion algebras A and B is a bijective map φ: A → B that preserves the norm. In other words, φ satisfies the following condition:

\widetilde{\mathrm{N}} \left(\varphi\left(x\right)\right) = \widetilde{\mathrm{N}} x

for all x in A.

Exercise III.4.(b) of Lam's Introduction to Quadratic Forms over Fields

The problem of determining the conditions of "norm" isomorphism in quaternion algebra is Exercise III.4.(b) of Lam's Introduction to Quadratic Forms over Fields. The problem statement is as follows:

If one defines $\widetilde{\mathrm{N}} x=\mathrm{T}\left(x^{2}\right)/ 2$ on $A=\left(\mathbb{R},i,j,k\right)$ , where $i^{2}=j^{2}=k^{2}=-1$ and $ij=k$ , $jk=i$ , $ki=j$ , show that the map $\varphi:A\rightarrow A$ defined by $\varphi\left(x\right)=\left(\mathrm{tr}\left(x\right),\mathrm{tr}\left(ix\right),\mathrm{tr}\left(jx\right),\mathrm{tr}\left(kx\right)\right)$ is an isomorphism.

Solution

To show that the map φ is an isomorphism, we need to verify that it preserves the norm. In other words, we need to show that:

\widetilde{\mathrm{N}} \left(\varphi\left(x\right)\right) = \widetilde{\mathrm{N}} x

for all x in A.

Using the definition of the norm, we have:

\widetilde{\{N}} \left(\varphi\left(x\right)\right) = \mathrm{T}\left(\varphi\left(x\right)^{2}\right)/2

Now, we can compute the square of φ(x):

\varphi\left(x\right)^{2} = \left(\mathrm{tr}\left(x\right),\mathrm{tr}\left(ix\right),\mathrm{tr}\left(jx\right),\mathrm{tr}\left(kx\right)\right)^{2}

= \left(\mathrm{tr}\left(x\right)^{2} + \mathrm{tr}\left(ix\right)^{2} + \mathrm{tr}\left(jx\right)^{2} + \mathrm{tr}\left(kx\right)^{2}\right)

+ 2\left(\mathrm{tr}\left(x\right)\mathrm{tr}\left(ix\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(kx\right)\right)

+ 2\left(\mathrm{tr}\left(ix\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(ix\right)\mathrm{tr}\left(kx\right) + \mathrm{tr}\left(jx\right)\mathrm{tr}\left(kx\right)\right)

Using the properties of the trace, we can simplify the expression:

\varphi\left(x\right)^{2} = \left(\mathrm{tr}\left(x\right)^{2} + \mathrm{tr}\left(ix\right)^{2} + \mathrm{tr}\left(jx\right)^{2} + \mathrm{tr}\left(kx\right)^{2}\right)

+ 2\left(\mathrm{tr}\left(x\right)\mathrm{tr}\left(ix\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(kx\right)\right)

+ 2\left(\mathrm{tr}\left(ix\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(ix\right)\mathrm{tr}\left(kx\right) + \mathrm{tr}\left(jx\right)\mathrm{tr}\left(kx\right)\right)

= \left(\mathrm{tr}\left(x\right)^{2} + \mathrm{tr}\left(ix\right)^{2} + \mathrm{tr}\left(jx\right)^{2} + \mathrm{tr}\left(kx\right)^{2}\right)

+ 2\left(\mathrm{tr}\left(x\right)\mathrm{tr}\left(ix\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(kx\right)\right)

+ 2\left(\mathrm{tr}\left(ix\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(ix\right)\mathrm{tr}\left(kx\right) + \mathrm{tr}\left(jxright)\mathrm{tr}\left(kx\right)\right)

= \left(\mathrm{tr}\left(x\right)^{2} + \mathrm{tr}\left(ix\right)^{2} + \mathrm{tr}\left(jx\right)^{2} + \mathrm{tr}\left(kx\right)^{2}\right)

+ 2\left(\mathrm{tr}\left(x\right)\mathrm{tr}\left(ix\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(kx\right)\right)

+ 2\left(\mathrm{tr}\left(ix\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(ix\right)\mathrm{tr}\left(kx\right) + \mathrm{tr}\left(jx\right)\mathrm{tr}\left(kx\right)\right)

= \left(\mathrm{tr}\left(x\right)^{2} + \mathrm{tr}\left(ix\right)^{2} + \mathrm{tr}\left(jx\right)^{2} + \mathrm{tr}\left(kx\right)^{2}\right)

+ 2\left(\mathrm{tr}\left(x\right)\mathrm{tr}\left(ix\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(kx\right)\right)

+ 2\left(\mathrm{tr}\left(ix\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(ix\right)\mathrm{tr}\left(kx\right) + \mathrm{tr}\left(jx\right)\mathrm{tr}\left(kx\right)\right)

= \left(\mathrm{tr}\left(x\right)^{2} + \mathrm{tr}\left(ix\right)^{2} + \mathrm{tr}\left(jx\right)^{2} + \mathrm{tr}\left(kx\right)^{2}\right)

+ 2\left(\mathrm{tr}\left(x\right)\mathrm{tr}\left(ix\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(jx\right) + \mathrm{tr}\left(x\right)\mathrm{tr}\left(kx\right)\right)

Q&A

Q: What is the norm of a quaternion?

A: The norm of a quaternion q is defined as: