How we apply representation theory to physics. I want to have a concrete idea of what people do with representation theory in physics. Here is what I think:
Corresponding to a specific "physics", there is particularly a Lie group (called G) of symmetry. Then we can build a framework by representing this Lie group as a group of linear transformations acting on a vector space V.


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*Each element of V would be a state of physics.

*Each element of the Lie algebra (corresponding to G) would be an observable (this is what I want to know if it's true or false for sure)


Then we can apply quantum concepts as eigenstate, eigenvalue, distribution,...
Am I wrong? If I am wrong, how can I be fixed?
(I just read accidentally about representation theory last week and I'm kind of excited about the idea of promoting a theory from a somewhat simple (fundamental) object as a group of symmetry)
 A: In quantum mechanics, states are vectors in (really lines in) Hilbert spaces $V$. Time evolution of states is described by the Hamiltonian $H$, which is some self-adjoint operator $H : V \to V$ such that the state $U(t) v$ after $t$ time units is given by
$$U(t) v = e^{iHt} v$$
(up to a constant involving Planck's constant that I'm ignoring); this is a restatement of the (time-independent) Schrödinger equation. See also Stone's theorem.
If a particular quantum system has symmetries given by a Lie group $G$, that Lie group acts unitarily (in nice cases; see Wigner's theorem) on $V$ in a way commuting with $H$, and (in nice cases) breaks it up into irreducible representations, for example if $G$ is compact, each of which is also an eigenspace for $H$. As a simple example, if $V = L^2(\mathbb{R}^3)$ describes the position of a particle in $\mathbb{R}^3$ under the influence of a spherically symmetric potential, then $G = \text{SO}(3)$ acts on $V$ preserving the eigenspaces of $H$. Now the elements of $\mathfrak{g}$ act by skew-adjoint operators, so multiplying or dividing them by $i$ gives self-adjoint operators which we can think of as observables corresponding to the action of $G$ by Noether's theorem. 
Wigner promoted the idea that types of particles should be thought of as irreducible representations of $G$, so understanding the irreducible decomposition of $H$ tells you what kind of particles occur in your theory, roughly speaking, with the observables coming from $\mathfrak{g}$ telling you various things about your particles, e.g. their momenta. In particular Wigner classified the irreducible representations of the Poincaré group to understand what kind of particles could show up in quantum field theory. Note that this group isn't compact. 
For further reading I recommend Singer's Linearity, Symmetry, and Prediction in the Hydrogen Atom, which thoroughly works out the details in the case of $L^2(\mathbb{R}^3)$ with $H$ coming from the Coulomb potential. The irreducible decomposition under the action of $\text{SO}(3)$ turns out to tell you about the structure of the periodic table, and in particular the periods in the periodic table turn out to be related to the dimensions of the irreducible representations of $\text{SO}(3)$. This is a very beautiful story and more people should know it. 
For shorter reading you might also be interested in this blog post where I describe a toy model of the above story on a finite graph. 
