Bresar, Lemma 1.3 - Finite Division Algebras .... real quaternions ....

In summary, Bresar, Lemma 1.3 is a fundamental result that states the necessary condition for the existence of finite division algebras, which have various applications in mathematics and physics. Real quaternions, a specific type of finite division algebra, are closely related to this lemma and have applications in computer graphics, robotics, and physics. The proof of the existence of real quaternions relies on Bresar, Lemma 1.3, and this lemma also has applications in abstract algebra, algebraic geometry, and number theory. The proof of Bresar, Lemma 1.3 involves using the concept of vector spaces and linear transformations over a field, and can be understood as showing that the dimension of a finite division algebra must
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I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...

I need help with some aspects of the proof of Lemma 1.3 ... ...

Lemma 1.3 reads as follows:
https://www.physicsforums.com/attachments/6209
In the above text by Matej Bresar we read the following:

" ... ... Set \(\displaystyle i := \frac{1}{ \sqrt{ -u^2 } } u , \ \ j := \frac{1}{ \sqrt{ -v^2 } } v\) , and \(\displaystyle k := ij.\)

It is straightforward to check that (1.1) holds ... ... "
I need some help in proving that \(\displaystyle ij = -ji = k\) ... sadly, I cannot get past substituting in the relevant formulas ... ...

Hope someone can help ...PeterEDIT ... I must admit that as I reflect further on Lemma 1.3 I am more confused than I first thought ... why is Bresar defining another 'multiplication' in V ... that is why define \circ ... and how does that definition play out in validating 1.1 ...

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In order for readers of the above post to appreciate the context of the post I am providing pages 1-3 of Bresar ... as follows ...
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I am familiar with the importance of understanding the foundations and proofs of mathematical concepts in order to fully grasp their implications in a scientific context. In this case, the topic of finite dimensional division algebras is crucial in many areas of science, including physics and engineering.

In regards to Lemma 1.3, I can offer some guidance on how to approach the proof. Firstly, it is important to understand the definitions of i, j, and k as provided in the text. These are defined as elements in the division algebra V, and the goal is to show that they satisfy the properties in (1.1). As you mentioned, it is helpful to substitute in the relevant formulas and work through the algebraic manipulations step by step.

To prove that ij = -ji = k, you can start by substituting in the definitions of i and j, and then using the properties of division algebras to simplify the expression. You can also use the fact that V is a division algebra to show that (1.1) holds for the elements i, j, and k. It may also be helpful to refer back to the definitions of division algebras and their properties as needed.

Regarding the use of the symbol \circ, this is just another way of representing the multiplication operation in V. It is defined in this way to make it easier to prove (1.1) and to show that the elements i, j, and k satisfy the properties of a division algebra.

I hope this helps in your understanding of Lemma 1.3. If you have any further questions or need more specific guidance, please do not hesitate to ask. As scientists, it is important for us to work through these proofs and understand the underlying concepts in order to fully appreciate the implications of finite dimensional division algebras in our research. Best of luck with your studies!
 

FAQ: Bresar, Lemma 1.3 - Finite Division Algebras .... real quaternions ....

What is the significance of Bresar, Lemma 1.3 in the study of finite division algebras?

Bresar, Lemma 1.3 is a fundamental result in the study of finite division algebras, which are algebraic structures that have a multiplication operation and a division operation defined for every element. This lemma states that any finite division algebra over a field must have its dimension as a power of the characteristic of the field. This result is important because it provides a necessary condition for the existence of finite division algebras, which have various applications in mathematics and physics.

Can you explain the concept of real quaternions and their relationship to finite division algebras?

A real quaternion is a type of finite division algebra that extends the real numbers by adding three imaginary units, denoted as i, j, and k, that satisfy certain multiplication rules. Real quaternions have applications in computer graphics, robotics, and physics. They are also closely related to the study of finite division algebras, as they are a specific example of such algebras.

How is Lemma 1.3 used in the proof of the existence of real quaternions?

The proof of the existence of real quaternions relies on Bresar, Lemma 1.3, as it provides a necessary condition for the existence of finite division algebras over a field. By showing that the real numbers can be extended to a finite division algebra of dimension 4, the existence of real quaternions is established.

Are there any other applications of Bresar, Lemma 1.3 in mathematics?

Yes, Bresar, Lemma 1.3 has applications in various areas of mathematics, such as abstract algebra, algebraic geometry, and number theory. It is also useful in the study of finite groups and finite fields.

Can you explain the proof of Bresar, Lemma 1.3 in simple terms?

The proof of Bresar, Lemma 1.3 involves using the concept of vector spaces and linear transformations over a field. It can be understood as follows: if a finite division algebra over a field exists, then its dimension must be a power of the characteristic of the field. This is because the multiplication operation in a finite division algebra can be represented as a linear transformation, and the dimension of the algebra is equal to the dimension of its vector space. Therefore, the dimension must be a power of the characteristic, as any linear transformation over a finite field has a dimension that is a power of the field's characteristic.

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