By Mark R. Sepanski
Blending algebra, research, and topology, the research of compact Lie teams is without doubt one of the most lovely components of arithmetic and a key stepping stone to the speculation of normal Lie teams. Assuming no previous wisdom of Lie teams, this booklet 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 class, and the Borel–Weil Theorem. the mandatory Lie algebra idea is additionally built within the textual content with a streamlined procedure targeting linear Lie groups.
• offers an technique that minimizes complicated prerequisites
• Self-contained and systematic exposition requiring no earlier publicity to Lie theory
• Advances fast 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 type and building of representations
• workouts sprinkled throughout
This starting graduate-level textual content, aimed basically at Lie teams classes and comparable issues, assumes familiarity with ordinary suggestions from staff idea, research, and manifold idea. scholars, learn mathematicians, and physicists attracted to Lie concept will locate this article very useful.
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Additional info for Compact Lie Groups
B) With respect to the spin representation action, show that (g · u, g · v) = (u, v) for u, v ∈ S = W and g ∈ Spinn (R). (c) For n even, show that (·, ·) restricts to a nondegenerate form on S ± = ± W when m is even, but restricts to zero when m is odd. 1 Constructing New Representations Given one or two representations, it is possible to form many new representations using standard constructions from linear algebra. For instance, if V and W are vector spaces, one can form new vector spaces via the direct sum, V ⊕W , the tensor product, V ⊗ W , or the set of linear maps from V to W , Hom(V, W ).
2. Let (π, V ) and (π , V ) be ﬁnite-dimensional representations of a Lie group G. (1) T ∈ Hom(V, V ) is called an intertwining operator or G-map if T ◦ π = π ◦ T . (2) The set of all G-maps is denoted by HomG (V, V ). (3) The representations V and V are equivalent, V ∼ = V , if there exists a bijective G-map from V to V . 2 Examples Let G be a Lie group. A representation of G on a ﬁnite-dimensional vector space V smoothly assigns to each g ∈ G an invertible linear transformation of V satisfying π(g)π(g ) = π(gg ) for all g, g ∈ G.
29 For n ≥ 3, show that the polynomial x12 + · · · + xn2 is irreducible over C. However, show that x12 + · · · + xn2 is a product of linear factors over Cn (R). 30 Let (·, ·) be any symmetric bilinear form on Rn or Cn . 26 by replacing x ⊗x +|x|2 by x ⊗x −(x, x) in the deﬁnition of I. 35 still holds. If (·, ·) has signature p, q on Rn , the resulting Clifford algebra is denoted C p,q (R) (so Cn (R) = C0,n (R)) and if (·, ·) is the negative dot product on Cn , the resulting Clifford algebra is denoted by Cn (C).