# Tensor product of tensor algebra

I am just learning about classical algebras and bialgebras and I'm now reading about the coproduct $\Delta':T(V)\to T(V)\bigotimes T(V)$. My question is about the algebra $T(V)\bigotimes T(V)$. Its elements are sums of elements that look like $$v_1v_2...v_n\bigotimes v_1'v_2'...v_m'$$ but as I have understood it $v_1v_2...v_n$ is just short for $$v_1\bigotimes v_2\bigotimes ... \bigotimes v_n$$ so could we write the element $v_1v_2...v_n\bigotimes v_1'v_2'...v_m'$ as $v_1v_2...v_nv_1'v_2'...v_m'$ or do we need to keep track of "which" tensor product is used where?

Thankful for any help=)

• The elements in the tensor product do not all look like that: they look (they are, in fact) sums of elements looking like that. Feb 4, 2017 at 20:42
• You are right, thanks! But can we omit the tensor symbol in each term then or no? Feb 4, 2017 at 20:45
• No, you can't. ${}$ Feb 4, 2017 at 20:49
• A useful notation, that should save you some space (or ink), is replacing $\bigotimes$ with $\otimes$, or even $\mid$, as in $v_1v_2\mid v_3v_4$.
– Pedro
Feb 6, 2017 at 18:02

You need to keep track of the two different tensor products. For example $v_1\otimes v_2v_3$ is different from $v_1v_2\otimes v_3$.