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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?

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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.

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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

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