Lie Groups (2nd Edition) (Graduate Texts in Mathematics, Volume 225)

By Daniel Bump

This ebook is meant for a one-year graduate direction on Lie teams and Lie algebras. The booklet is going past the illustration conception of compact Lie teams, that's the foundation of many texts, and gives a gently selected diversity of fabric to provide the coed the larger photograph. The e-book is equipped to permit assorted paths throughout the fabric counting on one's pursuits. This moment variation has giant new fabric, together with superior discussions of underlying ideas, streamlining of a few proofs, and plenty of effects and subject matters that weren't within the first edition.

For compact Lie teams, the e-book covers the Peter–Weyl theorem, Lie algebra, conjugacy of maximal tori, the Weyl workforce, roots and weights, Weyl personality formulation, the elemental staff and extra. The booklet maintains with the learn of complicated analytic teams and basic noncompact Lie teams, overlaying the Bruhat decomposition, Coxeter teams, flag kinds, symmetric areas, Satake diagrams, embeddings of Lie teams and spin. different issues which are taken care of are symmetric functionality thought, the illustration thought of the symmetric workforce, Frobenius–Schur duality and GL(n) × GL(m) duality with many functions together with a few in random matrix conception, branching principles, Toeplitz determinants, combinatorics of tableaux, Gelfand pairs, Hecke algebras, the "philosophy of cusp forms" and the cohomology of Grassmannians. An appendix introduces the reader to using Sage mathematical software program for Lie workforce computations.

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Evidence. word that SU(2) = enable G and H be topological teams. by way of an area homomorphism G −→ H we suggest the next info: an area U of the id and a continual map φ : U −→ H such that φ(uv) = φ(u)φ(v) each time u, v, and uv ∈ U . this suggests that φ(1G ) = 1H , so if u, u−1 ∈ U we've got φ(u−1 ) = φ(u). We could in addition exchange U through U ∩ U −1 so this can be real for all u ∈ U . thirteen The common disguise 87 Theorem thirteen. three. enable G and H be topological teams, and suppose that G is just attached. permit U be a local of the id in G. Then any neighborhood homomorphism U −→ H might be prolonged to a homomorphism G −→ H. evidence. permit g ∈ G. enable p : [0, 1] −→ G be a course with p(0) = 1G , p(1) = g. (Such a course exists simply because G is path-connected. ) We first express that there exists a special course q : [0, 1] −→ H such that q(0) = 1H , and q(v) q(u)−1 = φ p(v) p(u)−1 (13. 1) whilst u, v ∈ [0, 1] and |u − v| is sufficiently small. We be aware that once u and v are sufficiently shut, p(v)p(u)−1 ∈ U , so this is sensible. to build a course q with this estate, locate zero = x0 < x1 < · · · < xn = 1 such that once u and v lie in an period [xi−1 , xi+1 ], we've p(v)p(u)−1 ∈ U (1 i < n). outline q(x0 ) = 1H , and if v ∈ [xi , xi+1 ] outline q(v) = φ p(v) p(xi )−1 q(xi ). (13. 2) This definition is recursive simply because the following q(xi ) is outlined by means of (13. 2) with i changed via i − 1 if i > zero. With this definition, (13. 2) is admittedly actual for 1. certainly, if v ∈ [xi−1 , xi ] (the subinterval for which v ∈ [xi−1 , xi+1 ] if i this isn't a definition), now we have q(v) = φ p(v) p(xi−1 )−1 q(xi−1 ), so what we have to express is that q(xi ) q(xi−1 )−1 = φ p(v) p(xi )−1 −1 φ p(v)p(xi−1 )−1 . It follows from the truth that φ is a neighborhood homomorphism that the right-hand aspect is φ p(xi ) p(xi−1 )−1 . changing i by means of i − 1 in (13. 2) and taking v = xi , this equals q(xi )q(xi−1 )−1 . Now (13. 1) follows for this direction via noting that if ϵ = 12 min |xi+1 − xi |, then whilst |u − v| < ϵ, u, v ∈ [0, 1], there exists an i such that u, v ∈ [xi−1 , xi+1 ], and (13. 1) follows from (13. 2) and the truth that φ is a neighborhood homomorphism. This proves that the trail q exists. to teach that it really is precise, imagine that (13. 1) is legitimate for |u−v| < ϵ, and select the xi in order that |xi −xi+1 | < ϵ; then for v ∈ [xi , xi+1 ], (13. 2) is right, and the values of q are decided by means of this estate. subsequent we point out how you can convey that if p and p′ are path-homotopic, and if q and q ′ are the corresponding paths in H, then q(1) = q ′ (1). it truly is sufficient to end up this within the detailed case of a path-homotopy t → pt , the place p0 = p and p1 = p′ , such that there exists a series zero = x1 · · · xn = 1 with pt (u)pt′ (v)−1 ∈ U whilst u, v ∈ [xi−1 , xi+1 ] and t and t′ ∈ [0, 1]. For even though a common path-homotopy won't fulfill this assumption, it may be damaged into steps, every one of which does. therefore, we outline 88 thirteen The common hide qt (v) = φ pt (v) p(xi )−1 q(xi ) while v ∈ [xi , xi+1 ] and confirm that this qt satisfies qt (v)qt (u)−1 = φ pt (v) pt (u)−1 whilst |u − v| is small.

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