Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I was just curious in describing the notion that the tensor algebra of a vector space (That is the direct sum of all spaces containing k-tensors for each k) need not be commutative because I am having trouble coming up with an explicit example besides maybe multiplication of matrices.

Let $V\ $ be a finite dimensional vector space such that the dimension of $V\,$ is 2 or greater.

How do you show the tensor algebra $T(V) = \oplus_{k=1}^{\infty} T^k(V)$ is not commutative?

share|improve this question

1 Answer 1

up vote 7 down vote accepted

The tensor algebra is generated by formal products $v_1\otimes\cdots\otimes v_n$. If we permute any of the $v_i$, the resulting elementary tensor is not the same. This is a complete lack of commutativity.

share|improve this answer
3  
Oh, and you need the assumption about it's dimension so that you can find two linearly independent vectors $v_1$ and $v_2$ so that $v_1\otimes v_2$ is different from $v_2\otimes v_1$. –  Joe Johnson 126 Oct 21 '11 at 1:56
    
Apart from minor hazards of the notion of "formal product", one should indeed decide on a definite strategy for proving that $x\otimes y\not=y\otimes x$ for $x,y$ linearly independent. As @Joe Johnson 126's comment indicates, this is the crucial point. –  paul garrett Nov 8 '11 at 14:08

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.