Compact Lie Groups by Mark R. Sepanski

By Mark R. Sepanski

Blending algebra, research, and topology, the research of compact Lie teams is among the most pretty parts of arithmetic and a key stepping stone to the speculation of common Lie teams. Assuming no past wisdom of Lie teams, this publication covers the constitution and illustration concept of compact Lie teams. assurance comprises the development of the Spin teams, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Maximal Torus Theorem, the Commutator Theorem, the Weyl Integration and personality formulation, the top Weight type, and the Borel-Weil Theorem. The ebook develops the required Lie algebra conception with a streamlined process targeting linear Lie groups.

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42 2 Representations cos t − sin t cos t i sin t ∈ SU (2) and let ηt = ∈ SU (2). sin t cos t i sin t cos t Since H is SU (2) invariant, 12 (K t ± iηt ) z 1k0 z 2n−k0 ∈ H . Thus, when the limits exLet K t = ist, dtd 12 (K t ± iηt ) z 1k0 z 2n−k0 |t=0 ∈ H . 26) 1 d (K t ± iηt ) z 1k0 z 2n−k0 |t=0 = 2 dt for + k0 z 1k0 −1 z 2n−k0 +1 (k0 − n) z 1k0 +1 z 2n−k0 −1 for − . Induction therefore implies that Vn (C) ⊆ H , and so Vn (C) is irreducible. 3 for notation). Let Dm (Rn ) be the space of complex constant coefficient differential operators on Rn of degree m.

31. If V is a finite-dimensional S O(n)-invariant subspace of continuous functions on S n−1 , then V contains a nonzero S O(n − 1)-invariant function: Here the action of S O(n) on V is, as usual, given by (g f )(s) = f (g −1 s). Since S O(n) acts transitively on S n−1 and V is nonzero invariant, there exists f ∈ V , so f (1, 0, . . , 0) = 0. 44 2 Representations Define f (s) = S O(n−1) f (gs) dg. If { f i } is a basis of V , then f (gs) = g −1 f (s) and so may be written as f (gs) = i ci (g) f i (s) for some smooth functions ci .

Because r g ◦ rh = rhg , it follows that the modular function c : G → R\{0} is a homomorphism. 41). 41 imply that f (gh) dg = G ( f ◦ rh )ωG = c(h) G G = c(h) G ( f ◦ rh )(rh∗ ωG ) rh∗ ( f ωG ) = c(h) sgn (c(h)) f ωG = G f (g) dg. 42). We already know that ωG is the unique (up to ±1) left invariant normalized volume form on G. More generally, the corresponding measure dg is the unique left invariant normalized Borel measure on G. 47. For compact G, the measure dg is the unique left invariant Borel measure on G normalized so G has measure 1.

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