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 previous wisdom of Lie teams, this ebook covers the constitution and illustration thought of compact Lie teams. integrated 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 required Lie algebra idea can also be built within the textual content with a streamlined technique targeting linear Lie groups.

Key Features:

• presents an procedure that minimizes complicated prerequisites

• Self-contained and systematic exposition requiring no prior 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 speedy transition to the type and building of representations

• workouts sprinkled throughout

This starting graduate-level textual content, aimed basically at Lie teams classes and similar subject matters, assumes familiarity with user-friendly recommendations from staff thought, research, and manifold concept. scholars, study mathematicians, and physicists attracted to Lie conception will locate this article very useful.

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I z m ) | z k ∈ C} and W = {(z 1 , . . , z m , −i z 1 , . . , −i z m ) | z k ∈ C}. For n odd, take W = {(z 1 , . . , z m , i z 1 , . . , i z m , 0) | z k ∈ C}, W = {(z 1 , . . , z m , −i z 1 , . . , −i z m , 0) | z k ∈ C}, and e0 = (0, . . , 0, 1). Compared to our previous representations, the action of the spin representation is fairly complicated. We state the necessary deﬁnition below, although it will take some work to provide appropriate motivation and to show that everything is well deﬁned.

Xm ) and y = (y1 , . . , ym ) in Rm and u ∈ R, let (x, y, u) = (x1 , . . , xm , y1 , . . , ym , u) ∈ Rn . In particular, (x, y, u) = 12 (x −i y, i(x −i y), 0)+ 1 (x +i y, −i(x +i y), 0) + (0, 0, u). 35) 1 (x − i y) − 2ι(x + i y) + (−1)deg m iu . 36. For n even, the half-spin representations S ± of Spinn (R) are irreducible. For n odd, the spin representation S of Spinn (R) is irreducible. 3 Examples of Irreducibility 45 Proof. Using the standard basis {e j }nj=1 , calculate (e j ± ie j+m )(ek ± iek+m ) = e j ek ± i(e j ek+m + e j+m ek ) − e j+m ek+m for 1 ≤ j, k ≤ m.

Wm } of W and let {w1 , . . , (wi , w j ) is 0 when i = j and 1 when i = j. With respect to this basis, acts in a particularly simple fashion. If 1 ≤ i 1 < · · · < i k ≤ m, then (wi1 · · · wik wi1 · · · wik ) kills p W for p < k, maps 32 2 Representations k W onto Cwi1 ∧ · · · ∧ wik , and preserves p W for p > k. An inductive argument on n − k therefore shows that the image of contains each projection of W onto Cwi1 ∧ · · · ∧ wik . Successive use of the operators (wi ) and (w j ) can then be used to map wi1 ∧· · ·∧wik to any other w j1 ∧· · ·∧w jl .