Irreducible highest weight representations as a graded algebra

Question

Let $L$ be a semisimple Lie algebra and let $V(\lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $\lambda$. How can we view the sum \begin{align*} \oplus_{n\in\mathbb{N}}V(n\lambda) \end{align*} as a graded algebra with $n$th homogeneous component $V(n\lambda)$? More precisely, if $\lambda$ and $\mu$ are dominant weights, how do we define multiplication on $V(\lambda)V(\mu)$ and why $V(\lambda)V(\mu) \subset V(\lambda + \mu)$?

Answer 1

If $v_{\lambda}\in V(\lambda)$ and $v_{\mu}\in V(\mu)$ are highest weight vectors, then $v_{\lambda}\otimes v_{\mu}$ is s highest weight vector in $V(\lambda)\otimes V(\mu)$ of weight $\lambda+\mu$, which spans the weight space for that weight. Hence this spans an irreducible representation isomorphic to $V(\lambda+\mu)$ and projecting along an invariant complement, one a unique (up to scale) homomorphism $V(\lambda)\otimes V(\mu)\to V(\lambda+\mu)$. If you realize each $V(n\lambda)$ as a subrepresentation of $S^nV(\lambda)$ in this way, the choice of a $v_\lambda\in V(\lambda)$ gives you a highest weight vector in each $V(n\lambda)$ and thus a scale for all the homomorphisms in question. (Viewed in this way, one is actually constructing a subalgebra of the symmetric algebra $S^*V(\lambda)$ of $V(\lambda)$.