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Studyng the book Group Theory in Physics of 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 representaion of a Lie algebra? I know what is is a representation of a group, but I had never heard about a representation of a algebra before. I think the book doesn't explain it, and searching in the internet I can't find a direct answer. So, is there a relation between the algebra and the representation of a group?

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    $\begingroup$ I don't have enough information to really judge the book, but that sentence doesn't sound right. A rep. of the group is not a rep. of the algebra; at best, it induces one. Moreover, the converse is definitely not true: there are reps. of the algebra that do not lift to a rep. of the group. Maybe get a better book? $\endgroup$ – AccidentalFourierTransform Jun 1 at 14:09
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    $\begingroup$ Roughly: A rep of a group is a set of matrices that satisfy the group table upon matrix multiplication. A rep of an algebra is a set of matrices that satisfy the commutation relations of the algebra. $\endgroup$ – MBolin Jun 1 at 14:09
  • $\begingroup$ ...and completing @MBolin's summary above, exponentiation of the (Lie) algebra representation matrices produces the group representation matrices. (Up to mathematical fussing that book teaches you to bypass and gloss over on the way to physics.) By the way, if you are asking this question you might not be ready for that book: you are supposed to already know how to exponentiate Pauli matrices in your sleep. Getting an organized overlay of formalism and definitions before practical dexterity in using the tools of that trade almost always leads to confusion and grief... $\endgroup$ – Cosmas Zachos Jun 1 at 14:17
  • $\begingroup$ FWIW, half-integer spin representations of the Lie algebra $so(3)$ are not group representations of the Lie group $SO(3)$; only a projective representation thereof. $\endgroup$ – Qmechanic Jun 1 at 14:57
  • $\begingroup$ yes it looks like slight abuse of notation since strictly speaking a representation of the algebra is not a representation of the group. $\endgroup$ – ZeroTheHero Jun 1 at 15:00
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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

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