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Concept
Conformal group
Summary
In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.
Several specific conformal groups are particularly important:
The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V
For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of dilations.
The conformal group of the sphere is generated by the inversions in circles. This group is also known as the Möbius group.
In Euclidean space En, n > 2, the conformal group is generated by inversions in hyperspheres.
In a pseudo-Euclidean space Ep,q, the conformal group is Conf(p, q) ≃ O(p + 1, q + 1) / Z2.
All conformal groups are Lie groups.
In Euclidean geometry one can expect the standard circular angle to be characteristic, but in pseudo-Euclidean space there is also the hyperbolic angle. In the study of special relativity the various frames of reference, for varying velocity with respect to a rest frame, are related by rapidity, a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities. Thus, they are conformal transformations with respect to the hyperbolic angle.
A method to generate an appropriate conformal group is to mimic the steps of the Möbius group as the conformal group of the ordinary complex plane. Pseudo-Euclidean geometry is supported by alternative complex planes where points are split-complex numbers or dual numbers. Just as the Möbius group requires the Riemann sphere, a compact space, for a complete description, so the alternative complex planes require compactification for complete description of conformal mapping.
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