Free Algebras

**ilia1987** · Nov3-09, 03:24 PM

I currently self study from the book "A Course in Modern Mathematical Physics" by Peter Szekeres, and I'm currently reading the chapter on tensors, which he defines using the concept of Free Vector Spaces.
He gives a re-definition of Grassmann's algebras introduced in the previous section by using the concept of free algebra. And I had the following problem while reading the text:
(This is a Citation:) "
Let $LaTeX Code: \\mathcal{F}(V)$ be the free associative algebra over a real vector space

, and let $LaTeX Code: \\mathcal{S}$ be the ideal generated by all elements of $LaTeX Code: \\mathcal{F}(V)$ of the form $LaTeX Code: u \\otimes T \\otimes v + v \\otimes T \\otimes u$ where $LaTeX Code: u,v\\in V$ and $LaTeX Code: T\\in \\mathcal{F}(V)$ . The general element of $LaTeX Code: \\mathcal{S}$ is $LaTeX Code: S \\otimes u \\otimes T \\otimes v \\otimes U + S \\otimes v \\otimes T \\otimes u \\otimes U$ where $LaTeX Code: u,v\\in V$ and $LaTeX Code: S,T,U\\in \\mathcal{F}(V)$
"
My question is: Why can every element be expressed in this way?
An Ideal is first of all a Vector Subspace, right? So the sum of any two such "general elements" is supposed to be a general element too.
In other words, how do I prove that :
$LaTeX Code: S_1 \\otimes u_1 \\otimes T_1 \\otimes v_1 \\otimes U_1 + S_1 \\otimes v_1 \\otimes T_1 \\otimes u_1 \\otimes U_1 \\ \\ \\ \\ + \\ \\ \\ \\ S_2 \\otimes u_2 \\otimes T_2 \\otimes v_2 \\otimes U_2 + S_2 \\otimes v_2 \\otimes T_2 \\otimes u_2 \\otimes U_2 \\ \\ \\ \\ = \\ \\ \\ \\ S_3 \\otimes u_3 \\otimes T_1 \\otimes v_3 \\otimes U_3 + S_3 \\otimes v_3 \\otimes T_3 \\otimes u_3 \\otimes U_3$
?

Thank you for your time.

**HallsofIvy** · Nov4-09, 05:07 AM

Originally Posted by ilia1987

I currently self study from the book "A Course in Modern Mathematical Physics" by Peter Szekeres, and I'm currently reading the chapter on tensors, which he defines using the concept of Free Vector Spaces.
He gives a re-definition of Grassmann's algebras introduced in the previous section by using the concept of free algebra. And I had the following problem while reading the text:
(This is a Citation:) "
Let $LaTeX Code: \\mathcal{F}(V)$ be the free associative algebra over a real vector space , and let $LaTeX Code: \\mathcal{S}$ be the ideal generated by all elements of $LaTeX Code: \\mathcal{F}(V)$ of the form $LaTeX Code: u \\otimes T \\otimes v + v \\otimes T \\otimes u$ where $LaTeX Code: u,v\\in V$ and $LaTeX Code: T\\in \\mathcal{F}(V)$ . The general element of $LaTeX Code: \\mathcal{S}$ is $LaTeX Code: S \\otimes u \\otimes T \\otimes v \\otimes U + S \\otimes v \\otimes T \\otimes u \\otimes U$ where $LaTeX Code: u,v\\in V$ and $LaTeX Code: S,T,U\\in \\mathcal{F}(V)$
"
My question is: Why can every element be expressed in this way?

Because that is what "generated by" means!

An Ideal is first of all a Vector Subspace, right So the sum of any two such "general elements" is supposed to be a general element too.
In other words, how do I prove that :
$LaTeX Code: S_1 \\otimes u_1 \\otimes T_1 \\otimes v_1 \\otimes U_1 + S_1 \\otimes v_1 \\otimes T_1 \\otimes u_1 \\otimes U_1 \\ \\ \\ \\ + \\ \\ \\ \\ S_2 \\otimes u_2 \\otimes T_2 \\otimes v_2 \\otimes U_2 + S_2 \\otimes v_2 \\otimes T_2 \\otimes u_2 \\otimes U_2 \\ \\ \\ \\ = \\ \\ \\ \\ S_3 \\otimes u_3 \\otimes T_1 \\otimes v_3 \\otimes U_3 + S_3 \\otimes v_3 \\otimes T_3 \\otimes u_3 \\otimes U_3$
?

Thank you for your time.

**ilia1987** · Nov4-09, 11:04 AM

I'm sorry, but if that's what generated means, the general element should be of the form:

$LaTeX Code: \\forall \\ x\\in \\mathcal{S} , \\ x=\\displaystyle\\sum_r (S_r \\otimes v_r \\otimes T_r \\otimes u_r \\otimes U_r \\ + \\ S_r \\otimes u_r \\otimes T_r \\otimes v_r \\otimes U_r)$

where $LaTeX Code: S_r,T_r,U_r \\in \\mathcal{F}(V)$ and $LaTeX Code: u_r,v_r\\in V$ .
The question is whether any sum of this form can be expressed as a single element of the same form.