Lie groups and algebras

Sattinger, D. H., and Weaver, O. L. (1986), Lie Group and Algebras with Applications to Physics, Geometry and Mechanics, Springer Verlag, New York. 

Lie groups and Lie algebras are discussed in many textbooks (Hamermesh, 1962 Gilmore, 1974 Wyboume, 1974 Bamt and Raczka, 1986). We follow closely the notation of Wyboume (1974). There are also a number of mathematical texts (Miller, 1968 Talman, 1968 Vilenkin, 1968 Miller, 1977 Olver, 1986). 

In quantum mechanics our main interest is in the unirreps of Lie groups and these correspond to representations of the associated Lie algebras for... 

As we will see below, in modern problems of Hamiltonian mechanics and geometry there often appear systems which in general possess certain noncommutative symmetry groups. In this connection, we will need some known facts from the theory of Lie groups and Lie algebras. For the reader s convenience, we have collected all such facts in this section. We omit the proofs, of course, and refer the reader to the manuals (for instance, [28]). 

What does algebraization of any concrete mechanical system result in It allows us to apply the developed apparatus of the theory of Lie groups and Lie algebras. As is seen from the studies carried out in recent years (see, in particular, [89]-[94], [130] [149] etc.), this makes it possible to exhibit rather efficiently and in an explicit form the polynomial and rational integrals of many interesting dynamic systems. 

Gilmore, R. (1974), Lie Groups, Lie Algebras and Some of Their Applications, Wiley, New York. 

Like Lie groups. Lie algebras have representations. In this section we define and discuss these representations. In the examples we develop facility calculating with partial differential operators. Finally, we prove Schur s Lemma along with two propositions used to construct subrepresentations. 

In a sense that can be made quite precise. Lie groups are global objects and Lie algebras are local objects. To put it another way, Lie algebras are infinitesimal versions of Lie groups. In our main examples, the representation of the Lie group 50(3) on operates by rotations of functions, while the rep-... 

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. 

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