I am reading Lusztig's book Introduction to quantum groups. I have a question on page 3. In the fourth line of section 1.2.2, it is said that $'f \otimes 'f$ is associative. I don't know why.
I think that $((x_1\otimes x_2)(x'_1\otimes x'_2))(x''_1\otimes x''_2) = v^{|x_2||x'_1|}(x_1x'_1\otimes x_2x'_2) \otimes (x''_1\otimes x''_2)$ $= v^{|x_2||x'_1|+|x_2x'_2||x''_1|}x_1x'_1x''_1\otimes x_2x'_2x''_2$. But it seems that $(x_1\otimes x_2)((x'_1\otimes x'_2)(x''_1\otimes x''_2))$ does not equal this. Why $((x_1\otimes x_2)(x'_1\otimes x'_2))(x''_1\otimes x''_2) = (x_1\otimes x_2)((x'_1\otimes x'_2)(x''_1\otimes x''_2))$?