Tensor Algebras and Graded Algebras - Cooperstein

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In summary, the conversation discusses questions related to the text in Bruce N. Cooperstein's book on Advanced Linear Algebra, specifically Section 10.3 on Tensor Algebra. The questions involve understanding the multiplication involved in certain products, such as S(v_1) S(v_2) and \mathcal{A}_k \mathcal{A}_l. The nature of the multiplication depends on the specific algebra \mathcal{A} and its operations.
  • #1

Math Amateur

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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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Math Amateur said:
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].
 
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Thanks Steven ... appreciate your help ...
 

1. What is a tensor algebra?

A tensor algebra is a mathematical structure that allows for the manipulation and study of tensors, which are multi-dimensional arrays of numbers. It is a vector space generated by a given vector space, and includes all possible tensor products of vectors in that space.

2. How are tensor algebras used in physics?

Tensor algebras are used in physics to describe the relationships between physical quantities and their transformation properties. They are especially useful in the study of relativity and electromagnetism, as well as in quantum mechanics and field theory.

3. What is a graded algebra?

A graded algebra is a type of algebraic structure where the elements are divided into different "grades" or "degrees". This allows for a more general and flexible way of defining and manipulating mathematical objects, and has applications in areas such as algebraic geometry and representation theory.

4. How are tensor algebras and graded algebras related?

Tensor algebras are a special case of graded algebras, where the grading is given by the degree of the tensor. In other words, tensor algebras are a specific type of graded algebra that are used to study tensors, while graded algebras have a wider range of applications and uses.

5. Are there real-world applications of tensor algebras and graded algebras?

Yes, there are many real-world applications of both tensor algebras and graded algebras. In addition to their use in physics and mathematics, they also have applications in computer science, signal processing, and data analysis. For example, they can be used to analyze and manipulate complex data sets and to design efficient algorithms for processing large amounts of data.

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