Tensor Algebras and Graded Algebras - Cooperstein

  • #1
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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:


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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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  • #2
stevendaryl
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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)## ... ... ?
Well, the original assumption is that [itex]S[/itex] is a linear map from the vector space [itex]V[/itex] to the algebra [itex]\mathcal{A}[/itex]. So [itex]S(v)[/itex] is an element of the algebra [itex]\mathcal{A}[/itex], whatever that is--it's left unspecified. But every algebra over a field [itex]\mathbb{F}[/itex] has its own notion of multiplication and addition. So you're just using that notion of multiplication in interpreting [itex]S(v_1) S(v_2)[/itex]

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## ... ... ?
[itex]\mathcal{A}_k[/itex] and [itex]\mathcal{A}_l[/itex] are subsets of [itex]\mathcal{A}[/itex]. So I think that [itex]\mathcal{A}_k \mathcal{A}_l[/itex] just means the set of all elements [itex]x[/itex] such that [itex]x = y z[/itex], where [itex]y[/itex] is an element of [itex]\mathcal{A}_k[/itex] and [itex]z[/itex] is an element of [itex]\mathcal{A}_l[/itex].
 
  • #3
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Thanks Steven ... appreciate your help ...
 

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