T2: Representations of Lie groups and Lie algebras

Question 1

Define a representation (ρ, V ) of a (complex) Lie group G

Answer 1

The smooth (holomorphic) Lie group homomorphism
ρ : G → GL(V )

Question 2

How do we define the derivative of the action ρ?

Answer 2

Dρ: g → gl(V ) = End(V)

I.e. the Lie algebra action