Representations of Compact Lie Groups

Compact Lie groups are ‘perfect’ entities in modern mathematics because: (a) They are differentiable (and even analytical) manifolds of finite dimension and therefore can be analysed with the help of the most powerful tool of mathematics: ordinary and partial differential operators (equations); (b) Denoting by £(G) or L(G) = T e (G) the tangent space to G in unity e, we find that L(G) can be identified with the set of left-invariant vector fields on G (or with the set of appropriate differential operators of order one). Thus L(G) naturally becomes a Lie algebra: on L(G) there exists a structure [·, ·] which satisfies the axioms of Lie algebra (see below); (c) This rich algebraic structure, in turn, makes it possible to apply the powerful algebraic tools (theory of Lie algebras); (d) An abstract (finite dimensional) Lie algebra g defines (up to an isomorphism) a simply connected Lie group G such that £(G) = g. Therefore investigations of the Lie group G can be to large extent replaced by investigations of its Lie algebra g; (e) Compactness of a Lie group G makes it possible to use the theory of representations of compact groups developed by Weyl and Peter. This theory is extraordinarily beautiful and simple: it reduces to large extend to the spectral theory of compact operators; (f) And finally every complex, compact (connected) commutative Lie group is a torus. Moreover, as it was shown by E. Cartan, every compact Lie group contains a maximal torus T (any other such a torus is conjugated T 1 = gTg −1 ); and any point g 0 of the group G belongs to some maximal torus: G = U g gTg −1. This fundamental Cartan theorem makes it possible to apply to T (and thus to the whole of G) the theory of representations of abelian groups: every representation of the torus T decomposes into the finite (and orthogonal) sum of unit representations (over ℂ), the so called weights. Thus the knowledge of these weights is the fundamental element of the theory of representations of compact Lie groups.