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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=)

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    The elements in the tensor product do not all look like that: they look (they are, in fact) **sums** of elements looking like that. – 2017-02-04
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    You are right, thanks! But can we omit the tensor symbol in each term then or no? – 2017-02-04
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    No, you can't. ${}$ – 2017-02-04
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    A useful notation, that should save you some space (or ink), is replacing $\bigotimes$ with $\otimes$, or even $\mid$, as in $v_1v_2\mid v_3v_4$. – 2017-02-06

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