Fundamental Concepts of Higher Algebra by A. Adrian Albert

By A. Adrian Albert

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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.

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