Realization of a Lie algebra

Associated with each operator realization of a Lie algebra we generally have a vector space on which these operators act. For the realization given by L this might be either an abstract space of angular momentum states lm), 1 = 0, 1,... m = —/, — l + 1,..., l or a concrete realization of them as spherical harmonic functions Ylm(6, (j)). We can then consider the matrix elements of the operators with respect to this vector space of states and this leads to the important concept of a matrix representation of a Lie algebra. 

When the representation space is not precisely specified, we often talk about a realization of a Lie algebra rather than the representation (cf. also... 

The following general commutator identities are quite useful for simplifying the commutation relations, which occur in the determination of realizations of a Lie algebra. The well-known relations... 

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