Algebra, irreducible representation

The characters 4,1,0 form a reducible representation in the C3 point group and we require to reduce it to a set of irreducible representations, the sum of whose characters under each operation is equal to that of the reducible representation. We can express this algebraically as... 

The irreducible representations of may be used to define a set of elements of the group algebra called "projection operators . The projection operator associated with the / th irreducible representation is defined by... 

For each irreducible representation JW there are different projection operators, so in all there are 2 n — g such operators. As we shall see presently, they are linearly independent, and thus form an alternative basis for the group algebra any element of U can be represented as a linear combination of the... 

The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A. The quantum numbers are conveniently arranged in patterns (or tableaux), called Young tableaux. Tensor representations of Lie algebras are characterized by a set of integers... 

The irreducible representations of unitary algebras, U( ), are characterized by a set of n integers, corresponding to all possible partitions of an integer s,... 

The matrices of the irreducible representations provide one with a special set of group algebra elements. We define... 

Here, the connection or the gauge potential A assuming values in a(n irreducible) representation R of the compact, semisimple Lie algebra g of the Lie group G is of the form... 

It can be easily checked by direct computation that we have really obtained a realization of the Lie algebra g in a Hilbert (Fock) space, [T a, T fc] = ifabc fc, in accordance with (11), where Ta = T f/aL . For an irreducible representation R, the second-order Casimir operator C2 is proportional to the identity operator I, which, in turn, is equal to the number operator N in our Fock representation, that is, if T" —> Ta, then I /V 5/,/a . Thus we obtain an important for our further considerations constant of motion N ... 

Proposition 8.4 (Schur s Lemma) Suppose (fl, Vi, pf) and (fl, Pf) are irreducible representations of the Lie algebra fl. Suppose that T Vi V2 is a homomorphism of representations. Then there are only two possible cases ... 

In other words, the representations U of 5m(2) as differential operators on homogeneous polynomials in two variables are essentially the only finitedimensional irreducible representations, and they are classified by their dimensions. Unlike the Lie group 50(3), the Lie algebra sm(2) has infinitedimensional irreducible representations on complex scalar product spaces. See Exercise 8.10. 

The results of the current section, both the lowering operators and the classification, will come in handy in Section 8.4, where we classify the irreducible representations of so(4). One can apply the classification of the irreducible representations of the Lie algebra sm(2) to the study of intrinsic spin, as an alternative to our analysis of spin in Section 10.4. More generally, raising and lowering operators are widely useful in the study of Lie algebra representations. 

The Casimir operator is a useful tool for identifying a representation of the Lie algebra 5m(2). In this section we investigate Casimir operators and apply them to the classification of the finite-dimensional irreducible representations of the Lie algebra so 4 ). 

The next proposition classifies finite-dimensional irreducible representations of 50(4). Recall from Proposition 8.3 that 5o(4) = 5m(2) su(2), so the representations of the two Lie algebras must be identical. Hence it suffices to classify the finite-dimensional irreducible representations of 5m(2) 5m(2). 

Exercise 8.10 In this exercise we construct infinite-dimensional irreducible representations of the Lie algebra su (2). Suppose k. is a complex number such that L in for any nonnegative integer n. Consider a countable set S = vo, Vi, 172,... and let V denote the complex vector space of finite linear combinations of elements of S. Show that V can be made into a complex... 

The algebra of the Sp4 group coincides with the algebra of the rotation group of five-dimensional Euclidean space R5, i.e. these groups are locally isomorphic. The irreducible representations of the Sp4 group can be characterized by a set of two parameters (v, t) where the seniority quantum number v for the five-dimensional quasispin group indicates the number... 

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