Let $T$ be a split torus over a field $k$. Then the dual torus $\hat{T}$ is defined to be the unique torus such that $$ X_*(T)=X^*(\hat{T}), $$ where the left hand side is the cocharacter lattice of $T$ and the right hand side is the character lattice of $\hat{T}$.
- Is there any way to concretely write down $\hat{T}$ in terms of $T$?
- Is there any explicit isomorphism between $X_*(T)$ and $X^*(\hat{T})$?
