By Mark R. Sepanski
Blending algebra, research, and topology, the examine of compact Lie teams is likely one of the most pretty parts of arithmetic and a key stepping stone to the idea of normal Lie teams. Assuming no past wisdom of Lie teams, this publication covers the constitution and illustration concept of compact Lie teams. incorporated is 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 required Lie algebra idea is additionally constructed within the textual content with a streamlined method concentrating on linear Lie groups.
• presents an strategy that minimizes complex prerequisites
• Self-contained and systematic exposition requiring no past publicity to Lie theory
• Advances speedy to the Peter–Weyl Theorem and its corresponding Fourier theory
• Streamlined Lie algebra dialogue reduces the differential geometry prerequisite and permits a extra quick transition to the class and building of representations
• workouts sprinkled throughout
This starting graduate-level textual content, aimed basically at Lie teams classes and comparable themes, assumes familiarity with uncomplicated options from crew idea, research, and manifold conception. scholars, examine mathematicians, and physicists drawn to Lie thought will locate this article very useful.
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Extra info for Compact Lie Groups
4) (g · Q) (u) = (−bu + a)n Q au + b −bu + a a −b ∈ SU (2), Q ∈ Vn , and u ∈ C. 1) b a this yields a representation of SU (2). In fact, this apparently new representation is old news since it turns out that Vn ∼ = Vn (C2 ). To see this, we need to construct a bijective intertwining operator from Vn (C2 ) to Vn . Let T : Vn (C2 ) → Vn be given by (T P)(u) = P(u, 1) for P ∈ Vn (C2 ) and u ∈ C. This map is clearly bijective. To see that T is a G-map, use the deﬁnitions to calculate that for g = [T (g · P)] (u) = (g · P) (u, 1) = P(au + b, −bu + a) = (−bu + a)n P au + b ,1 −bu + a = (−bu + a)n (T P) (u) = [g · (T P)] (u), so T (g · P) = g · (T P) as desired.
A representation of a Lie group G on a ﬁnite-dimensional complex vector space V is a homomorphism of Lie groups π : G → G L(V ). The dimension of a representation is dim V . Technically, a representation should be denoted by the pair (π, V ). When no ambiguity exists, it is customary to relax this requirement by referring to a representation (π, V ) as simply π or as V . Some synonyms for expressing the fact that (π, V ) is a representation of G include the phrases V is a G-module or G acts on V .
Show that 0 1 the left invariant measure is x −2 d xd y but the right invariant measure is x −1 d xd y. 44 Let G be a Lie group and ϕ : U → V ⊆ Rn a chart of G with e ∈ U , 0 ∈ V , and ϕ(e) = 0. Suppose f is any integrable function on G supported in U . (a) For x ∈ V , write g = g(x) = ϕ −1 (x) ∈ U . Show the function l x = ϕ ◦ l g−1 ◦ ϕ −1 is well deﬁned on a neighborhood of x. , ∂l∂ xx |x = |det J |, where the Jacobian matrix J is given by ∂(l ) Ji, j = ∂ xx i j |x . Pull back the relation ωg = l g∗−1 ωe to show that the left invariant measure dg can be scaled so that ( f ◦ ϕ −1 )(x) f dg = G V ∂l x |x d x 1 .