Conformal group

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

  ${\displaystyle Q(Tx)=\lambda ^{2}Q(x)}$

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 E^p,q, the conformal group is Conf(p, q) ≃ O(p + 1, q + 1) / Z₂.

All conformal groups are Lie groups.