Definition of the tensor product of representations

Question

I'm a bit confused about the following definition:

Let $\rho_1:G \to Aut(V_1)$, $\rho: G \to Aut(V_2) $ be two representations of the same group $G$. Then a tensor product of representations is defined as: $$ \rho_1 \otimes\rho_2:G \to Aut(V_1 \otimes V_2)\\ g \mapsto (\rho_1\otimes\rho_2)(g):= \rho_1(g)\otimes \rho_2(g) $$

Question: Isn't object $\rho_1(g)\otimes \rho_2(g)$ belongs to something like $Aut(V_1)\otimes Aut(V_2)$ (since $\rho_i(g) \in Aut(V_i)$), and not to $Aut(V_1 \otimes V_2)$?. But, honestly, then I don't understand if the tensor product of two non-abelian group is even defined.

Answer 1

If $A:V_1\to V_1$ and $B:V_2\to V_2$ are linear transformations, then $A\otimes B$ is defined to be the unique linear transformation $V_1\otimes V_2\to V_1\otimes V_2$ that satisfies $(A\otimes B)(x\otimes y)=A(x)\otimes B(y)$.

Answer 2

Given an element $$ x = \sum_{k} a_k v_{1, k} \otimes v_{2, k} \in V_1 \otimes V_2, $$ the action of $\rho_1 (g) \otimes \rho_2 (g)$ on $x$ is: $$(\rho_1 (g) \otimes \rho_2 (g))( x) = \sum_{k} a_k \rho_1 (g) ( v_{1, k}) \otimes \rho_2 (g) (v_{2, k}).$$

Thus $\rho_1 (g) \otimes \rho_2 (g)$ is a linear map from $V_1 \otimes V_2$ to $V_1 \otimes V_2$. So $\rho_1 (g) \otimes \rho_2 (g)$ is indeed in ${\rm Aut}(V_1 \otimes V_2)$.