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Non-homomorphic Isomorphism – Exercise
The following is Exercise 2.4(a) of Topics in Tensor Categories by Etingof, Nikshych, Ostrik.
(a) Give an example of a finite dimensional vector space $V$ and a subspace $W$ such that $V \otimes V \cong W \oplus W \oplus W$, but $V \otimes V
cong W \oplus W$.
I have attempted this without success. Here is what I have attempted.
Let $V = \text{span}\{v_1, v_2\}$. Let $W = \text{span}\{w_1, w_2\}$. My guess is that I may be able to show $\text{span}\{v_1\otimes w_1, v_2\otimes w_1, v_1\otimes w_2, v_2\otimes w_2\}$ is a basis for $V \otimes V$, and hence that $V \otimes V \cong W \oplus W \oplus W$. If this is true, we have that $\text{dim}(V \otimes V) = 4$ and $\text{dim}(W) = 2$, so $V \otimes V
cong W \oplus W$.
EDIT: I was incorrect above. It is true that $\{v_1\otimes w_1, v_2\otimes w_1, v_1\otimes w_2, v_2\otimes w_2\}$ is a basis for $V \otimes V$. For example, from the spanning property, $w_1 \otimes w_1 + w_2\otimes w_1 + w_1 \otimes w_2 + w_2 \otimes w_2 \in \text{span}\{v_1\otimes w_1, v_2\otimes w_1, v_1\otimes w_2, v_2\otimes w_2\}$. Since $w_1 \otimes w_1 + w_2\
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