## Compact Lie Groups by Mark R. Sepanski

By Mark R. Sepanski

Blending algebra, research, and topology, the research of compact Lie teams is likely one of the most pretty parts of arithmetic and a key stepping stone to the speculation of basic Lie teams. Assuming no past wisdom of Lie teams, this booklet covers the constitution and illustration idea of compact Lie teams. insurance 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 class, and the Borel-Weil Theorem. The booklet develops the mandatory Lie algebra idea with a streamlined strategy concentrating on linear Lie groups.

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The contents of this publication were used in classes given by way of the writer. the 1st used to be a one-semester direction for seniors on the collage of British Columbia; it was once transparent that stable undergraduates have been completely in a position to dealing with uncomplicated workforce conception and its program to easy quantum chemical difficulties.

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11), although the changes will not affect the ﬁnite-dimensional case. Two representations will be called equivalent if they are the same up to, basically, a change of basis. Recall that Hom(V, V ) is the set of all linear maps from V to V . 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, this deﬁnes a representation. As ﬁne and natural as this representation is, it actually contains a smaller, even nicer, representation. Write = ∂x21 + · · · + ∂x2n for the Laplacian on Rn . 5). 5. , Hm (Rn ) = {P ∈ Vm (Rn ) | P = 0}. If P ∈ Hm (Rn ) and g ∈ O(n), then (g · P) = g · ( P) = 0 so that g · P ∈ Hm (Rn ). In particular, the action of O(n) on Vm (Rn ) descends to a representation of O(n) (or S O(n), of course) on Hm (Rn ). It will turn out that these representations do not break into any smaller pieces.

44 to show that the invariant integral on S O(3) is given by f (g) dg = S O(3) 1 8π 2 π 2π 0 0 2π f (α(θ )β(φ)α(ψ)) sin φ dθdφdψ 0 for integrable f on S O(3). 48 Let θ α(θ) = ei 2 0 θ 0 e−i 2 and β (θ) = cos θ2 − sin θ2 sin θ2 cos θ2 . 47, show that the invariant integral on SU (2) is given by SU (2) f (g) dg = 1 8π 2 for integrable f on SU (2). π 2π 0 0 2π 0 f (α(θ )β(φ)α(ψ)) sin φ dθdφdψ 2 Representations Lie groups are often the abstract embodiment of symmetry. However, most frequently they manifest themselves through an action on a vector space which will be called a representation.