| Peter-weyl Theorem |
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Information AboutPeter-weyl Theorem |
| CATEGORIES ABOUT PETER–WEYL THEOREM | |
| harmonic analysis | |
| representation theory of topological groups | |
| mathematical theorems | |
| fourier analysis | |
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To state the theorem we need first the idea of the Hilbert Space over ''G'', ''L''2(''G''); this makes sense because Haar Measure exists on ''G''. Calling it ''H'', the group ''G'' has a unitary Representation on ''H'' by Acting on the left, or on the right. This implies a representation of ''G''×''G'' (via ρ((''h'',''k'')) {Link without Title} (''g'') = ''f''(''h''−1''gk'')). This representation decomposes into the sum of for each finite Irreducible Unitary Representation of ''G'' where is the Dual Representation . That is, there is a Direct Sum description of ''H'' with the indexation by all the classes (up to isomorphism) of irreducible unitary representations of ''G''. This implies immediately the structure of ''H'' for the left or right representations of ''G'', which comes out as a direct sum of each ρ as many times as its dimension (always finite). Structure of compact topological groups From the theorem, one can deduce a significant general structure theorem. Let ''G'' be a compact topological group, which we assume Hausdorff . For any finite-dimensional ''G''-invariant subspace ''V'' in ''L''2(''G''), where ''G'' Acts on the left, we consider the image of ''G'' in GL(''V''). It is closed, since ''G'' is compact, and a subgroup of the Lie Group GL(''V''). It follows by a Basic Theorem (of Élie Cartan ) that the image of ''G'' is a Lie group also. If we now take the limit (in the sense of . |
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