By A. Adrian Albert
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This is often the lengthy awaited follow-up to Lie Algebras, half I which lined an enormous a part of the speculation of Kac-Moody algebras, stressing basically their mathematical constitution. half II bargains often with the representations and functions of Lie Algebras and includes many move references to half I. The theoretical half principally bargains with the illustration conception of Lie algebras with a triangular decomposition, of which Kac-Moody algebras and the Virasoro algebra are best examples.
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E. a vector space with three endomorphisms X, Y, Z, satisfying commutation relations XY − Y X = Z, Y Z − ZY = X, ZX − XZ = Y . 49 any such group can be written as G = G/Z for ˜ and a discrete central subgroup Z ⊂ G. Thus, representations of some simply-connected group G ˜ satisfying ρ(Z) = id. 1). 4. 51. Then any complex representation of g has a unique structure of representation of gC , and Homg (V, W ) = HomgC (V, W ). In other words, categories of complex representations of g, gC are equivalent.
M. 1) (f1 , f2 ) = f1 (g)f2 (g) dg. 38). 2) δik δjl dim V Proof. The proof is based on the following easy lemma. 40. (1) Let V , W be non-isomorphic irreducible representations of G and f a linear map V → W . Then G gf g −1 dg = 0. (2) If f is a linear map V → V , then gf g −1 dg = tr(f ) dim V id. 7. Orthogonality of characters and Peter-Weyl theorem 49 Indeed, let f˜ = G gf g −1 dg. Then f˜ commutes with action of g: hf˜h−1 = G (hg)f (hg)−1 dg = f˜. By Schur lemma, f˜ = 0 for W = V and f˜ = λ id for W = V.
Then ϕ is an immersion, and ϕ(G1 ) is an immersed subgroup in G2 . In particular, every oneparameter subgroup in G is an immersed subgroup. One easily sees that immersed subgroups have the following properties. ˜ → G be an immersed subgroup. 42. Let i : H ˜ is a Lie group, and i is a morphism of Lie groups (1) H ˜ is a Lie subalgebra in g (it will be denoted by Lie(H)). (2) h = i∗ (T1 H) ˜ is closed in G, then H is a Lie subgroup. (3) If H = i(H) It turns out that this generalization solves all our problems.