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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 Theorem 1.4 ... ...
Theorem 1.4 reads as follows:
View attachment 6223Questions 1(a) and 1(b)
In the above text by Matej Bresar we read the following:
" ... ... Suppose $$n \gt 4$$. Let $$i, j, k$$ be the elements from Lemma 1.3.
Since the dimension of $$V$$ is $$n - 1$$, there exists $$v \in V$$ not lying in the linear span of $$i, j, k$$.
Therefore $$e := v + \frac{i \circ v}{2} i + \frac{j \circ v}{2} j + \frac{k \circ v}{2} k$$
is a nonzero element in $$V$$ and it satisfies $$i \circ e = j \circ e = k \circ e = 0$$ ... ... "
My questions are as follows:
(1a) Can someone please explain exactly why $$e := v + \frac{i \circ v}{2} i + \frac{j \circ v}{2} j + \frac{k \circ v}{2} k$$ is a nonzero element in $$V$$?
(1b) ... ... and further, can someone please show how $$e := v + \frac{i \circ v}{2} i + \frac{j \circ v}{2} j + \frac{k \circ v}{2} k$$ satisfies $$i \circ e = j \circ e = k \circ e = 0$$?Question 2
In the above text by Matej Bresar we read the following:
" ... ... However, from the first two identities we conclude $$eij = -iej = ije$$, which contradicts the third identity since $$ij = k$$ ... ... "I must confess Bresar has lost me here ... I'm not even sure what identities he is referring to ... but anyway, can someone please explain why/how we can conclude that $$eij = -iej = ije$$ and, further, how this contradicts $$ij = $$k?
Hope someone can help ...Help will be appreciated ... ...
PeterThe above post refers to Lemma 1.3.
Lemma 1.3 reads as follows:View attachment 6224
=====================================================
In order for readers of the above post to appreciate the context of the post I am providing pages 1-4 of Bresar ... as follows ...View attachment 6225
https://www.physicsforums.com/attachments/6226
View attachment 6227
View attachment 6228
I need help with some aspects of the proof of Theorem 1.4 ... ...
Theorem 1.4 reads as follows:
View attachment 6223Questions 1(a) and 1(b)
In the above text by Matej Bresar we read the following:
" ... ... Suppose $$n \gt 4$$. Let $$i, j, k$$ be the elements from Lemma 1.3.
Since the dimension of $$V$$ is $$n - 1$$, there exists $$v \in V$$ not lying in the linear span of $$i, j, k$$.
Therefore $$e := v + \frac{i \circ v}{2} i + \frac{j \circ v}{2} j + \frac{k \circ v}{2} k$$
is a nonzero element in $$V$$ and it satisfies $$i \circ e = j \circ e = k \circ e = 0$$ ... ... "
My questions are as follows:
(1a) Can someone please explain exactly why $$e := v + \frac{i \circ v}{2} i + \frac{j \circ v}{2} j + \frac{k \circ v}{2} k$$ is a nonzero element in $$V$$?
(1b) ... ... and further, can someone please show how $$e := v + \frac{i \circ v}{2} i + \frac{j \circ v}{2} j + \frac{k \circ v}{2} k$$ satisfies $$i \circ e = j \circ e = k \circ e = 0$$?Question 2
In the above text by Matej Bresar we read the following:
" ... ... However, from the first two identities we conclude $$eij = -iej = ije$$, which contradicts the third identity since $$ij = k$$ ... ... "I must confess Bresar has lost me here ... I'm not even sure what identities he is referring to ... but anyway, can someone please explain why/how we can conclude that $$eij = -iej = ije$$ and, further, how this contradicts $$ij = $$k?
Hope someone can help ...Help will be appreciated ... ...
PeterThe above post refers to Lemma 1.3.
Lemma 1.3 reads as follows:View attachment 6224
=====================================================
In order for readers of the above post to appreciate the context of the post I am providing pages 1-4 of Bresar ... as follows ...View attachment 6225
https://www.physicsforums.com/attachments/6226
View attachment 6227
View attachment 6228
Last edited: