Problem 7 - Algebra

Question

Let \(\mathrm{H}\) be the division ring over the reals generated by elements \(i, j, k\) such that \(i^{2}=j^{2}=k^{2}=-1\), and $$ i j=-j i=k, \quad j k=-k j=i, \quad k i=-i k=j $$ Then \(\mathrm{H}\) has an automorphism of order 2 , given by $$ a_{0}+a_{1} i+a_{2} j+a_{3} k \mapsto a_{0}-a_{1} i-a_{2} j-a_{3} k $$ Denote this automorphism by \(\alpha \mapsto \bar{\alpha}\). What is \(\alpha \bar{x}\) ? Show that the theory of hermitian forms can be carried out over \(\mathbf{H}\), which is called the division ring of quaternions (or by abuse of language, the non-commutative field of quaternions).

Step-by-Step Solution

Answer

Given a quaternion α = \(a_0 + a_1i + a_2j + a_3k\), the conjugate ᾱ = \(a_0 - a_1i - a_2j - a_3k\). Their product αᾱ = \(a_0^2 + a_1^2 + a_2^2 + a_3^2\), which is a real number. We define a hermitian form on H as \(Q(\alpha) = \alpha \alphā\). The hermitian form satisfies the properties \(Q(\lambda \alpha) = \lambda \lambdā Q(\alpha)\) and \(Q(\alpha+\beta) = Q(\alpha) + Q(\beta) - Q(\alpha - \beta)\). Thus, the theory of hermitian forms can be carried out over the quaternion division ring H.

Step 1: Understand the problem and notation

To begin, let's clarify the notations used in the problem. The quaternion field H is generated by 1, i, j, and k with some defining relations among them. The automorphism is a function that maps each element α in H to its conjugate ᾱ by negating coefficients of i, j, and k.

Step 2: Compute αᾱ

Given a quaternion α = \(a_0 + a_1i + a_2j + a_3k\), let's compute the product αᾱ. ᾱ = \(a_0 - a_1i - a_2j - a_3k\) We now calculate the product αᾱ: αᾱ = \((a_0 + a_1i + a_2j + a_3k) (a_0 - a_1i - a_2j - a_3k)\) By expanding the terms and using the properties of the quaternions, we have: αᾱ = \(a_0^2 + a_1^2 + a_2^2 + a_3^2\) (since the cross terms cancel out) Notice that the result is a real number (coefficient of 1 in the expansion).

Step 3: Show that the theory of hermitian forms can be carried out

To show that the theory of hermitian forms can be carried out over the quaternion division ring H, we first need to find a suitable definition for a hermitian form in H. A hermitian form is a quadratic form that is invariant under quaternion conjugation. In the context of quaternions, we can define a hermitian form as follows: \(Q(\alpha) = \alpha \alphā\), where α is an element of H. Using the result from Step 2, we can see that \(Q(\alpha) = a_0^2 + a_1^2 + a_2^2 + a_3^2\), which is a real number. Now, we can prove that Q satisfies some properties required for hermitian forms. Specifically, the following properties must hold true: 1. \(Q(\lambda \alpha) = \lambda \lambdā Q(\alpha)\) for any α in H and λ in ℝ. 2. \(Q(\alpha+\beta) = Q(\alpha) + Q(\beta) - Q(\alpha - \beta)\) for any α, β in H. Let's prove these properties: 1. \(Q(\lambda \alpha) = (\lambda \alpha)(\lambda \alpha)̄ = \lambda \alpha \alphā \lambdā = (\lambda \lambdā) (\alpha \alphā) = \lambda \lambdā Q(\alpha)\). 2. \(Q(\alpha+\beta) = (\alpha+\beta)(\alpha+\beta)̄ = (\alpha+\beta)(\alphā+\betā) = \alpha \alphā + \alpha \betā + \beta \alphā + \beta \betā = Q(\alpha) + Q(\beta) - Q(\alpha - \beta)\). Since both properties hold true, the theory of hermitian forms can indeed be carried out over the quaternion division ring H.

Key Concepts

Division Ring

The concept of a division ring is crucial in understanding quaternions, like the division ring H generated by elements \(i, j, k\). A division ring is similar to a field, as it is a ring where division is possible. However, unlike fields, a division ring may not be commutative (i.e., the multiplication operation can depend on the order of the elements). This is key when dealing with quaternions.
A division ring has the following properties:

• Every non-zero element has a multiplicative inverse.
• The associative property holds for multiplication.
• The multiplication in a division ring is not necessarily commutative.

Quaternions, therefore, are considered a division ring because they allow division, but the multiplication of quaternion elements is not the same in reverse order.

Automorphism

An automorphism is an important concept in understanding the structure of mathematical objects like quaternions. It refers to a bijective map from a mathematical object to itself that preserves the structure of that object. In the context of quaternions, the automorphism in question is a conjugation operation.
The specific automorphism described takes a quaternion \(a_0 + a_1 i + a_2 j + a_3 k\) and maps it to its conjugate \(a_0 - a_1 i - a_2 j - a_3 k\). This is a classic example of how transformations can be used to maintain structure within a division ring, specifically maintaining the norm or magnitude of the quaternion.
Key features of automorphisms include:

• Structure-preserving: They keep relationships between elements consistent.
• Bijection: They are both one-to-one and onto, meaning every element maps uniquely and covers the entire set.
• Symmetry: In some cases, like in quaternion conjugation, they reveal symmetrical properties of the object.

Recognizing how this automorphism results in a real number after multiplication shows the stabilizing effect of such operations.

Hermitian Forms

Hermitian forms are an extension of quadratic forms that can be used over non-commutative fields like quaternions. They provide a way to study properties of vectors across these kinds of algebraic structures. Hermitian forms are significant due to their properties of being invariant under conjugation.
In the quaternion context, the form \(Q(\alpha) = \alpha \bar{\alpha}\) results in real numbers, even though quaternions themselves can include complex components. This indicates that hermitian forms focus on capturing scalar quantities from non-commutative entities.
Hermitian forms fulfill certain characteristics:

• Invariance under conjugation, a critical feature to retain while dealing with quaternionic forms.
• Satisfying specific algebraic properties, like \(Q(\lambda \alpha) = \lambda \bar{\lambda} Q(\alpha)\) for any real number \(\lambda\).
• Decomposition property, meaning \(Q(\alpha + \beta) = Q(\alpha) + Q(\beta) - Q(\alpha - \beta)\).

These properties help carry out operations within the quaternion division ring, supporting complex calculations and analysis.

Non-commutative Algebra

Non-commutative algebra is a branch of algebra dealing with systems where the commutative property of multiplication does not hold. Quaternions are a classical example of non-commutative algebraic structures since for them \(ij = -ji\), illustrating order matters when multiplying elements.
In non-commutative settings, the regular operations and transformations can be significantly more complex. However, they provide deeper insights into phenomena that commutative algebras cannot always describe fully.
Key aspects of non-commutative algebra include:

• Lack of commutativity in multiplication, setting them apart from traditional algebraic structures.
• Rich structure, offering more layers of operation and transformation analysis.
• Widely applicable in different fields, such as for understanding rotations in physics.

The concepts learned here help capture the complexity and uniqueness of structures where operations like multiplication depend on element ordering, which is essential for advancing algebraic theories and their applications.