Relationship between representations of Lie groups and Lie algebras

Question

Consider all the finite-dimensional irreducible representations of a group.

1. For each finite-dimensional irreducible representation of a group, is there one and only one corresponding representation of the Lie algebra?

2. For a Lie group, are there always finite-dimensional irreducible representations of $n \times n$ matrices for all values of $n$?

3. For a Lie group, can there only be one finite-dimensional irreducible representation of $n \times n$ matrices for each value of n?

Answer 1

1. Every representation of a Lie group $G$ naturally induces a corresponding representation of the corresponding Lie algebra $\mathfrak{g}$, by differentiation. I'm not sure how to interpret "only one" in this question.

2. No. For example, $SO(3)$ has irreducible representations only in odd dimensions $1, 3, 5, \dots$.

3. No. For example, $SU(2) \times SU(2)$ has two nonisomorphic irreducible $2$-dimensional representations, one for each of its factors.