In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, an extension of Galois theory to the case of continuous symmetry groups was one of Lies principal motivations. Lie groups are smooth manifolds and as such can be studied using differential calculus. Lie groups play an role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various geometries by specifying an appropriate transformation group that leaves certain geometric properties invariant and this idea later led to the notion of a G-structure, where G is a Lie group of local symmetries of a manifold. On a global level, whenever a Lie group acts on an object, such as a Riemannian or a symplectic manifold. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry, Linear actions of Lie groups are especially important, and are studied in representation theory. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, a real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ, G × G → G μ = x y means that μ is a mapping of the product manifold G×G into G. These two requirements can be combined to the requirement that the mapping ↦ x −1 y be a smooth mapping of the product manifold into G. The 2×2 real invertible matrices form a group under multiplication, denoted by GL or by GL2 and this is a four-dimensional noncompact real Lie group. This group is disconnected, it has two connected components corresponding to the positive and negative values of the determinant, the rotation matrices form a subgroup of GL, denoted by SO. It is a Lie group in its own right, specifically, using the rotation angle φ as a parameter, this group can be parametrized as follows, SO =. Addition of the angles corresponds to multiplication of the elements of SO, thus both multiplication and inversion are differentiable maps. The orthogonal group also forms an example of a Lie group. All of the examples of Lie groups fall within the class of classical groups. Hilberts fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples, if the underlying manifold is allowed to be infinite-dimensional, then one arrives at the notion of an infinite-dimensional Lie group
Image: Irrational line on a torus
Image: Exponential Map 01
The circle of center 0 and radius 1 in the complex plane is a Lie group with complex multiplication.