A natural homomorphism between Lie representations Suppose that $\mathfrak{g}$ is a semisimple Lie algebra with Cartan subalgebra $\eta$ and $R$ is the sets of roots. Let $L(\lambda_1)$ and $L(\lambda_2)$ be finite dimensional irreducible highest weight modules with highest weights $\lambda_1$ and $\lambda_2$. Then I want to find a natural $\mathfrak{g}-$module homomorphism from $L(\lambda_1)\otimes L(\lambda_2)$ to $L(\lambda_1+\lambda_2)$. 
Suppose $u$ and $v$ are the primitive vectors respectively. By the definition it is easy to see that $u\otimes v$ is the primitive vector of $L(\lambda_1+\lambda_2)$. I have tried the maps $g_1u\otimes g_2v\mapsto g_1g_2(u\otimes v)$, $g_1u\otimes g_2v\mapsto g_1g_2(u\otimes v)+g_2g_1(u\otimes v)$ and $g_1u\otimes g_2v\mapsto g_1g_2(u\otimes v)-g_2g_1(u\otimes v)$. But all of them fail because the condition $\phi(g(g_1u\otimes g_2v))=g\phi(g_1u\otimes g_2v)$ does not hold. How can I define this homomorphism naturally?
Moreover, how can I prove that the space of the $\mathfrak{g}-$modules homomorphisms is of one dimension i.e. all the homomorphisms are the multiplications of the natural one with a constant.
 A: Note that elements of $\mathfrak{g}$ act on the tensor product of $\mathfrak{g}$-modules $V\otimes W$ by the formula
$$x.(v\otimes w)=(xv)\otimes w + v\otimes(xw).$$
With this in mind, not that if $v_i\in L(\lambda_i)$ ($i=1,2$) is a highest weight vector and $x\in \mathfrak{n}^+$, then 
$$x.(v_1\otimes v_2)=(xv_1)\otimes v_2+v_1\otimes(xv_2)=0\otimes v_2+v_1\otimes 0=0.$$
Therefore, $v_1\otimes v_2$ is a highest weight vector. Moreover, if $h\in \mathfrak{h}(=\eta?)$, then a calculation as above shows that
$$h(v_1\otimes v_2)=(\lambda_1+\lambda_2)(v_1\otimes v_2)$$
so $v_1\otimes v_2$ has weight $\lambda_1+\lambda_2$.
Now, since $L(\lambda_1)\otimes L(\lambda_2)$ is completely reducible, it follows that $U(\mathfrak{g})(v_1\otimes v_2)$ is isomorphic to $L(\lambda_1+\lambda_2)$, so take your map to be the composition
$$L(\lambda_1)\otimes L(\lambda_2)\twoheadrightarrow U(\mathfrak{g})(v_1\otimes v_2)\cong L(\lambda_1+\lambda_2),$$
where the first map is the natural projection and the second map is the isomorphism defined by the mapping $v_1\otimes v_2\mapsto w$ (here $w$ is any nonzero highest weight vector in $L(\lambda_1+\lambda_2)$).
A: I am not sure whether by "find" you mean "prove the existence of" or "find an explicit description". The latter problem will be hard in general. The simple homomorphism available in this situation is the one $L(\lambda_1+\lambda_2)\to L(\lambda_1)\otimes L(\lambda_2)$. In the language used in your question, this is given by (the linear extension of) $g\cdot v\mapsto g\cdot v_1\otimes v_2+v_1\otimes g\cdot v_2$ similarly as described in the answer of @DavidHill . This is just the embedding of the irreducible component of maximal highest weight into the tensor product. 
Existence of a homomorphism in the opposite direction follows from the fact that the invariant subspace formed by that irreducible component admits an invariant complement, so there exists an invariant projection onto it. This projection is complicated in general. To write it out, you need all the projections to other irreducible components (these usually are not hard to get, they are like traces) but also the way to re-embed these other irreducible components, and these are rather complicated unless the representaitons involved are really simple.  
