Representation and Lie algebra of $SO(3)$ Studying the book "Group Theory in Physics" by Wu-Ki Tung, I have read:

"... every representation of the [$SO(3)$] group is automatically a representation of the corresponding Lie algebra, (...) a representation of the Lie algebra will also provide us with a representation of the group".

But, what is exactly a representation of a Lie algebra? I know what is a representation of a group, but I had never heard about a representation of an algebra before. I think the book doesn't explain it, and searching on the internet I can't find a direct answer. So, is there a relation between the algebra and the representation of a group?
 A: Wu Ki Tung's writing is not too precise mathematically, it is true. "Group Theory in Physics" is the title, and the level of mathematical rigor is not the highest, thus this sloppy text, actually sloppy two-letter word: "Since the basis elements of the Lie algebra are generators of infinitesimal rotations, it is quite obvious that every representation of the group is automatically a representation of the corresponding Lie algebra". 
Representations of Lie groups induce representations of Lie algebras, in the exact sense specified by Theorem IV of page 408 of the 2nd volume of Cornwell's "Group Theory in Physics" (abbreviated by me to the purpose of this text): 

Let $\Gamma_{\mathcal{G}}$ be a d-dimensional analytic representation of linear Lie group $\mathcal{G}$, whose corresponding Lie algebra is $\mathcal{L}$. Then there exists a d-dimensional representation $\Gamma_{\mathcal{L}}$ of $\mathcal{L}$ defined for each $a\in\mathcal{L}$ by
  $$\Gamma_{\mathcal{L}}(a) = \left[\frac{d}{dt}\Gamma_{\mathcal{G}}(\exp (ta))\right]_{t=0}$$ 

This is a relation between a representation of the group and a representation of its (real) algebra. Generalizations to complex group and complex representations of their Lie algebra follow easily. 
The exact definition of a representation of a Lie algebra is here: https://en.wikipedia.org/wiki/Lie_algebra_representation
