I am reading a paper which begin by a reminder about root system associated to a simple lie algebra $\mathfrak g$. let $\mathfrak h\subset \mathfrak g$ a cartan subalgebra.
Question 1: It says that for any dominant weight $\lambda\in\mathfrak h$, we associate a unique up to isomorphism a simple $\mathfrak g-$module, denoted $V_\lambda$ , how to define $V_\lambda$ and why it is unique?
Question 2: it says that $V_\lambda$ contains a highest weight vector, but then evry dominant weight is a highest weight.? is the highest weight unique?
Thanks.