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Since right Haar measure is well-defined up to a positive scaling factor, this equation shows the modular function is independent of the choice of right Haar measure in the above equation. The modular function is a continuous group homomorphism from G to the multiplicative group of positive real numbers .
Any compact group is locally compact.. In particular the circle group T of complex numbers of unit modulus under multiplication is compact, and therefore locally compact. The circle group historically served as the first topologically nontrivial group to also have the property of local compactness, and as such motivated the search for the more general theory, presented here.
The representations of the group are found by considering representations of (), the Lie algebra of SU(2).Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation; [1] we will give an explicit construction of the representations at the group level below.
If G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called a Haar measure.Using the Haar measure, one can define a convolution operation on the space C c (G) of complex-valued continuous functions on G with compact support; C c (G) can then be given any of various norms and the completion will be a group algebra.
which would say the left translate of a right Haar measure is a multiple of a left Haar measure. However, the left translate of a right Haar measure is also a right Haar measure so what you're looking for is an assertion that a right Haar measure is a multiple of a left Haar measure, e.g. is itself left Haar. That's only true for unimodular groups.
Measurable acting groups are found in the intersection of measure theory and group theory, two sub-disciplines of mathematics. Measurable acting groups are the basis for the study of invariant measures in abstract settings, most famously the Haar measure , and the study of stationary random measures .
The notion of local compactness is important in the study of topological groups mainly because every Hausdorff locally compact group G carries natural measures called the Haar measures which allow one to integrate measurable functions defined on G. The Lebesgue measure on the real line is a special case of this.
A maximal torus in the special unitary group SU(n) ⊂ U(n) is just the intersection of T and SU(n) which is a torus of dimension n − 1. A maximal torus in the special orthogonal group SO(2 n ) is given by the set of all simultaneous rotations in any fixed choice of n pairwise orthogonal planes (i.e., two dimensional vector spaces).