Tensor Algebras and Graded Algebras - Cooperstein - Theorem 10.11 and Defn 10.7

In summary, Theorem 10.11 and Definition 10.7 in Section 10.3 of Bruce N. Cooperstein's book "Advanced Linear Algebra (Second Edition)" discuss the concept of tensor algebra and $\mathbb{Z}$-graded algebras. The multiplication involved in the product in the algebra $\mathcal{A}$ is $R$-bilinear and commutes with $\mathcal{A}$, while the multiplication in the product set $\mathcal{A}_k\mathcal{A}_l$ is simply a "product set".
  • #1
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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:View attachment 5565
View attachment 5566My 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 \(\displaystyle S^k \ : \ V^k \longrightarrow \mathcal{A}\) by

\(\displaystyle S^k (v_1, \ ... \ ... \ , v_k ) = S(v_1) S(v_2) \ ... \ ... \ S(v_k)\)

... ... ... "


My question is ... what is the form and nature of the multiplication involved between the elements in the product \(\displaystyle S(v_1) S(v_2) \ ... \ ... \ S(v_k)\) ... ... ?
Question 2

In the above text from Cooperstein in Definition 10.7 we read the following:

" ... ... An algebra \(\displaystyle \mathcal{A}\) is said to be \(\displaystyle \mathbb{Z}\)-graded if it is the internal direct sum of subspaces \(\displaystyle \mathcal{A}_k , k \in \mathbb{Z}\) such that

\(\displaystyle \mathcal{A}_k \mathcal{A}_l \subset \mathcal{A}_{k + l}\)

... ... ... "My question is ... what is the form and nature of the multiplication involved between the elements in the product \(\displaystyle \mathcal{A}_k \mathcal{A}_l\) ... ... ?Hope someone can help ...

Peter
 
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  • #2
Peter said:
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 \(\displaystyle S^k \ : \ V^k \longrightarrow \mathcal{A}\) by

\(\displaystyle S^k (v_1, \ ... \ ... \ , v_k ) = S(v_1) S(v_2) \ ... \ ... \ S(v_k)\)

... ... ... "


My question is ... what is the form and nature of the multiplication involved between the elements in the product \(\displaystyle S(v_1) S(v_2) \ ... \ ... \ S(v_k)\) ... ... ?

It is the product in the algebra $\mathcal{A}$ (recall an algebra has a "dual nature" as both an $R$-module *and* a ring, such that the ring multiplication is *compatible* with the module addition, AND the ring-action of $R$-this is why we require $R$-bilinearity. In addition, we require that $R$ commute with $\mathcal{A}$-that is, that $R \subseteq Z(\mathcal{A})$, the center of $\mathcal{A}$).
Question 2

In the above text from Cooperstein in Definition 10.7 we read the following:

" ... ... An algebra \(\displaystyle \mathcal{A}\) is said to be \(\displaystyle \mathbb{Z}\)-graded if it is the internal direct sum of subspaces \(\displaystyle \mathcal{A}_k , k \in \mathbb{Z}\) such that

\(\displaystyle \mathcal{A}_k \mathcal{A}_l \subset \mathcal{A}_{k + l}\)

... ... ... "My question is ... what is the form and nature of the multiplication involved between the elements in the product \(\displaystyle \mathcal{A}_k \mathcal{A}_l\) ... ... ?Hope someone can help ...

Peter

The set $\mathcal{A}_k\mathcal{A}_l = \{a \in \mathcal{A}: a = a_ka_l, a_k \in \mathcal{A}_k,a_l \in \mathcal{A}_l\}$ (it's just a "product set").
 
  • #3
Deveno said:
It is the product in the algebra $\mathcal{A}$ (recall an algebra has a "dual nature" as both an $R$-module *and* a ring, such that the ring multiplication is *compatible* with the module addition, AND the ring-action of $R$-this is why we require $R$-bilinearity. In addition, we require that $R$ commute with $\mathcal{A}$-that is, that $R \subseteq Z(\mathcal{A})$, the center of $\mathcal{A}$).The set $\mathcal{A}_k\mathcal{A}_l = \{a \in \mathcal{A}: a = a_ka_l, a_k \in \mathcal{A}_k,a_l \in \mathcal{A}_l\}$ (it's just a "product set").
Thanks Deveno ... appreciate the clarification ... most helpful ...

Peter
 

1. What is the significance of Theorem 10.11 in Tensor Algebras and Graded Algebras?

Theorem 10.11, also known as the Cooperstein Theorem, states that any algebra over a field can be embedded into a graded algebra over the same field. This is significant because it allows us to study more complex algebras by breaking them down into simpler graded algebras.

2. Can you explain the concept of tensor algebras?

Tensor algebras are mathematical structures that are used to represent and manipulate tensors, which are multi-dimensional arrays of numbers or symbols. They are important in fields such as physics and engineering, where tensors are used to describe physical quantities and transformations.

3. How is Theorem 10.11 related to Defn 10.7 in Tensor Algebras and Graded Algebras?

Defn 10.7 defines the concept of a graded algebra, which is a type of algebra that is decomposed into homogeneous subspaces. Theorem 10.11 shows that every algebra can be embedded into a graded algebra, providing a useful tool for studying and understanding the properties of algebras.

4. What are some applications of Tensor Algebras and Graded Algebras?

Tensor algebras and graded algebras have a wide range of applications in mathematics, physics, and engineering. They are commonly used in quantum mechanics, differential geometry, and representation theory, among other areas.

5. Are there any limitations to Theorem 10.11 and Defn 10.7 in Tensor Algebras and Graded Algebras?

Theorem 10.11 and Defn 10.7 have some limitations, as they only apply to algebras over fields. They also do not provide a unique way of embedding an algebra into a graded algebra, so different embeddings may result in different properties and structures. Additionally, the proof of Theorem 10.11 can be quite technical and may require advanced mathematical knowledge.

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