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Lie groups

Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a group which is such that the group operations are defined by smooth maps.

A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a compact Lie group is the circle: the group operation is simply rotation. This group, known as U (1) {\displaystyle \operatorname {U} (1)}, can be also characterised as the group of complex numbers of modulus 1 with multiplication as the group operation.

Other examples of Lie groups include special groups of matrices, which are all subgroups of the general linear group, the group of n × n {\displaystyle n\times n} matrices with non-zero determinant. If the matrix entries are real numbers, this will be an n 2 {\displaystyle n^{2}} -dimensional disconnected manifold. The orthogonal groups, the symmetry groups of the sphere and hyperspheres, are n (n − 1) / 2 {\displaystyle n(n-1)/2} dimensional manifolds, where n − 1 {\displaystyle n-1} is the dimension of the sphere. Further examples can be found in the table of Lie groups.