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Main Question or Discussion Point
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get an understanding of an aspect of Example 10.11 and Definition 10.7 in Section 10.3 ...
The relevant text in Cooperstein is as follows:
My questions related to the above text from Cooperstein are simple and related ... they are as follows:
Question 1
In the above text from Cooperstein ... at the start of the proof of Theorem 10.11 we read the following:
" ... ... Define a map ##S^k \ : \ V^k \longrightarrow \mathcal{A}## by
##S^k (v_1, \ ... \ ... \ , v_k ) = S(v_1) S(v_2) \ ... \ ... \ S(v_k)##
... ... ... "
My question is ... what is the form and nature of the multiplication involved between the elements in the product ## S(v_1) S(v_2) \ ... \ ... \ S(v_k)## ... ... ?
Question 2
In the above text from Cooperstein in Definition 10.7 we read the following:
" ... ... An algebra ##\mathcal{A}## is said to be ##\mathbb{Z}##graded if it is the internal direct sum of subspaces ##\mathcal{A}_k , k \in \mathbb{Z}## such that
##\mathcal{A}_k \mathcal{A}_l \subset \mathcal{A}_{k + l}##
... ... ... "
My question is ... what is the form and nature of the multiplication involved between the elements in the product ## \mathcal{A}_k \mathcal{A}_l## ... ... ?
Hope someone can help ...
Peter
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get an understanding of an aspect of Example 10.11 and Definition 10.7 in Section 10.3 ...
The relevant text in Cooperstein is as follows:
My questions related to the above text from Cooperstein are simple and related ... they are as follows:
Question 1
In the above text from Cooperstein ... at the start of the proof of Theorem 10.11 we read the following:
" ... ... Define a map ##S^k \ : \ V^k \longrightarrow \mathcal{A}## by
##S^k (v_1, \ ... \ ... \ , v_k ) = S(v_1) S(v_2) \ ... \ ... \ S(v_k)##
... ... ... "
My question is ... what is the form and nature of the multiplication involved between the elements in the product ## S(v_1) S(v_2) \ ... \ ... \ S(v_k)## ... ... ?
Question 2
In the above text from Cooperstein in Definition 10.7 we read the following:
" ... ... An algebra ##\mathcal{A}## is said to be ##\mathbb{Z}##graded if it is the internal direct sum of subspaces ##\mathcal{A}_k , k \in \mathbb{Z}## such that
##\mathcal{A}_k \mathcal{A}_l \subset \mathcal{A}_{k + l}##
... ... ... "
My question is ... what is the form and nature of the multiplication involved between the elements in the product ## \mathcal{A}_k \mathcal{A}_l## ... ... ?
Hope someone can help ...
Peter
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