I Tensor Algebras and Graded Algebras - Cooperstein

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

 

Thanks Steven ... appreciate your help ...