8.5 For further reading
Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · group representation · Lie group
The classification of representations is a standard topic in all textbooks that deal with Lie group representations. A good example is [40], which covers this material well, and from which the discussion here of the construction of representations as homogeneous polynomials is drawn (see pages 77-79). The calculation of the and the derivation of expressions for spherical harmonics as Lie algebra representations of appears in most quantum mechanics textbooks in one form or another (for example, see chapter 12 of [81]). Another source used here for the explicit constructions of representations is [20], chapters 27-30.
A conventional topic in books on representation theory in physics is that of the representation theory of the group , or even of for arbitrary The case is of great historical importance, because of its use in the classification and study of strongly interacting particles, The success of these methods is now understood as due to an approximate symmetry of the strong interaction theory corresponding to the existence of three diferent light quarks. The highest weight theory of representations can be generalized to the case of , as well as to finite dimensional representations of other Lie groups. We will not try and cover this topic here since it is a bit intricate, and is already very well-described in many textbooks aimed at mathematicians (e.g., [42]) and at physicists (e.g., [32]).
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