Help on Bresar's Left & Right Multiplication Maps on Algebras, Lemma 1.24

In summary: But since b_n \ne 0, we can solve for b_n and get b_n = \frac{1}{w_j}z_j. This means that b_n is invertible, which is a contradiction. Therefore, all the b_i must be equal to 0.In summary, Bresar's statement and proof of Lemma 1.24 are crucial in understanding the Wedderburn-Artin theorem. The assumption that A is a central simple algebra allows us to use the induction hypothesis in the proof, and the findings of c_n = 1 and n \gt 1 are necessary to establish that all
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...

I need some further help with the statement and proof of Lemma 1.24 ...

Lemma 1.24 reads as follows:View attachment 6258
My further questions regarding Bresar's statement and proof of Lemma 1.24 are as follows:Question 1

In the statement of Lemma 1.24 we read the following:

" ... ... Let \(\displaystyle A\) be a central simple algebra. ... ... "I am assuming that since \(\displaystyle A\) is central, it is unital ... that is there exists \(\displaystyle 1_A \in A\) such that \(\displaystyle x.1_A = 1_A.x = 1\) for all \(\displaystyle x \in A\) ... ... is that correct ... ?

Question 2

In the proof of Lemma 1.24 we read the following:

" ... ... Suppose \(\displaystyle b_n \ne 0\). ... ... "I am assuming that that the assumption \(\displaystyle b_n \ne 0\) implies that we are also assuming

that \(\displaystyle b_1 = b_2 = \ ... \ ... \ = b_{n-1} = 0\) ... ...

Is that correct?

Question 3

In the proof of Lemma 1.24 we read the following:

" ... ...where \(\displaystyle c_i = \sum_{ j = 1 }^m w_j b_i z_j\) ; thus \(\displaystyle c_n = 1\) for some \(\displaystyle w_j, z_j \in A\) ... ...

This clearly implies that \(\displaystyle n \gt 1\). ... ... "My question is ... why/how exactly must \(\displaystyle n \gt 1\) ... ?

Further ... and even more puzzling ... what is the relevance to the proof of the statements that

\(\displaystyle c_n = 1\) and \(\displaystyle n \gt 1\) ... ?

Why do we need these findings to establish that all the \(\displaystyle b_i = 0\) ... ?Hope someone can help ...

Peter

===========================================================*** NOTE ***

So that readers of the above post will be able to understand the context and notation of the post ... I am providing Bresar's first two pages on Multiplication Algebras ... ... as follows:View attachment 6259
View attachment 6260
 
Last edited:
Physics news on Phys.org
  • #2


Introduction to Noncommutative Algebra by Matej Bresar is a comprehensive book that covers various topics in noncommutative algebra. In Chapter 1, Bresar focuses on finite dimensional division algebras.

Lemma 1.24 is an important result in this chapter and provides a key step in proving the Wedderburn-Artin theorem. The lemma states:

Lemma 1.24: Let A be a central simple algebra. If A is finite dimensional, then A is a division algebra.

In order to understand the statement and proof of this lemma, you have raised some questions which I will address below.

Question 1:

In the statement of Lemma 1.24, A is assumed to be a central simple algebra. This means that A is a finite dimensional algebra with a center that coincides with the field of scalars. In other words, A is a central algebra and every element of A can be written as a linear combination of elements from the field of scalars. This implies that A is unital, since the field of scalars contains the identity element 1_A. Therefore, your assumption is correct.

Question 2:

In the proof of Lemma 1.24, Bresar assumes that b_n \ne 0. This assumption is made in order to apply the induction hypothesis. The induction hypothesis states that if a central simple algebra A is finite dimensional and has a nonzero element, then A is a division algebra. So, by assuming that b_n \ne 0, Bresar is able to use the induction hypothesis to prove that A is a division algebra.

Question 3:

In the proof of Lemma 1.24, Bresar defines c_i = \sum_{ j = 1 }^m w_j b_i z_j and shows that c_n = 1 for some w_j, z_j \in A. The reason why n \gt 1 is because if n = 1, then c_1 = w_1 b_1 z_1 = 1. This would imply that b_1 is invertible, which contradicts our assumption that b_1 = 0. Therefore, n \gt 1.

The relevance of c_n = 1 and n \gt 1 in the proof is that it allows us to conclude that all the b_i = 0. This is because if c_n = 1 and n \gt 1, then we can write
 

FAQ: Help on Bresar's Left & Right Multiplication Maps on Algebras, Lemma 1.24

1. What is the purpose of Lemma 1.24 in Bresar's Left & Right Multiplication Maps on Algebras?

Lemma 1.24 in Bresar's Left & Right Multiplication Maps on Algebras serves as a key step in proving the main theorem of the paper. It establishes the existence of left and right multiplication maps on algebras and their properties.

2. Why is it important to study left and right multiplication maps on algebras?

Left and right multiplication maps on algebras play a crucial role in understanding the structure and properties of algebras. They provide useful tools for studying the algebraic operations and their interactions within the algebra.

3. What does Lemma 1.24 state?

Lemma 1.24 states that for any algebra A, there exist left and right multiplication maps on A that satisfy certain properties. These maps are defined as linear transformations from A to A, and they preserve the algebraic operations of addition and multiplication.

4. How does Lemma 1.24 contribute to the overall understanding of algebras?

Lemma 1.24 is a fundamental result that allows for a deeper understanding of the structure and behavior of algebras. It provides a framework for studying the properties and interactions of algebraic operations within an algebra, leading to further developments in the field.

5. Are there any real-world applications of left and right multiplication maps on algebras?

Yes, left and right multiplication maps on algebras have various applications in areas such as coding theory, cryptography, and quantum computing. They also have important implications in the study of Lie algebras and their representations in physics and engineering.

Back
Top