Group Theory

Basics

Representation: p: G -> GL(V), a map from the group the set of linear maps from V to V (ambigous if Representation means the map p or the resulting set of linear maps)
Lie Algebra: Vector space + inner product (vector product) that is bilinear, follows the jacobi identity, and has x*x = 0

Generator operators U(R) = exp(-ihJ), J: operator / infinitesimal rotation
SO(3) has exactly one irrep of dim 1,3,5,... (=2j+1) -> How does this relate to number of eigenstates?
SU(2) allows even dim as well (j = half-integer) because SU(2) is double cover of SO(3) -> research
Wigner D-Matrix gives transition probabilities between eigenstates
Eigenstates form basis when discrete
When j= 3, m=-3,-2,-1,0,1,2,3 -> states |3m> are allowed -> real states are superpositions of these basis states / eigenstates -> 7-dim space needed to describe them (even though angular momentum needs only 3d classically) -> remember in general that degrees of freedom explode as 2^n because superpositions are allowed
j fixes total angular momentum, m fixes z component
angular momentum operator algebra is inherent to so(3) lie algebra

Where does the inherent need for SO(3) come from? Why are the possible state spaces the irreps of SO(3)? -> The operators that rotate states (in physical space) can be called U(R(theta)), where R in SO(3). There is a homomorphism between {U(R)} and SO(3), since U(R1R2)=U(R1)U(R2). Thus U(R) is a representation. Enforcing finite-dim and irrep, this fixes dim = 2j+1.
What about that L/J algebra? See "Lie Algebra" german wikipedia article -> Beispiele -> Aus der Physik -> Seems as though the relation does not only hold for those three "basis matrices", but all matrices, as long as your in the basis of the L_i (wtf even is a basis of matrices)
Is there an intuitive notion to see how angular momentum emerges from the shapes of the wave functions?
How does the emergence of azimuthal dependence in superpositions of (not azimuth-constraining) eigenstates play out mathematically?
Why is SU(2) the double cover of SO(3)? What does that mean?
What role do Pauli matrices play (generators of SO(2)? idk)
How does one come up with the idea that operators and eigenvalues/vectors are the appropriate mathematical tools to describe QM? How does it arise? BC this alone imposes a lot of the quantization maths, since eigenvalues are discrete
relation between Lie algebras and Lie groups not exactly clear
Whats the algebra of exp of matrices -> https://de.wikipedia.org/wiki/Matrixexponential
Is the generator exp(-ihJ) for SO(3) just the most convinient choice or the only choice?

Character tables and relation between conjugacy classes and irreps
Further sheets
Further lectures, explicitly everything after lecture 09
Wigner-Eckart Theorem (in lecture, sheet and indip research)
- Clebsch Gordan coeff as seperate research?