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Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.
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 : (,) = means that μ is a smooth mapping of the product manifold G × G into G. The two requirements can be combined to the single requirement ...
See Table of Lie groups for a list. General linear group, special linear group. SL 2 (R) SL 2 (C) Unitary group, special unitary group. SU(2) SU(3) Orthogonal group, special orthogonal group. Rotation group SO(3) SO(8) Generalized orthogonal group, generalized special orthogonal group. The special unitary group SU(1,1) is the unit sphere in the ...
There is an unfortunate conflict between the notations for the alternating groups A n and the groups of Lie type A n (q). Some authors use various different fonts for A n to distinguish them. In particular, in this article we make the distinction by setting the alternating groups A n in Roman font and the Lie-type groups A n (q) in italic.
Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a universal cover whose center is the fundamental group of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the ...
3. simply laced group (a simple Lie group is simply laced when its Dynkin diagram is without multiple edges). 4. simple root. A subset of a root system is called a set of simple roots if it satisfies the following conditions: is a linear basis of .
Lie group decomposition; Lie group–Lie algebra correspondence; Lie groupoid; Lie point symmetry; Lie product formula; Lie–Palais theorem; Lie's third theorem;
Every connected Lie group is isomorphic to its universal cover modulo a discrete central subgroup. [34] So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the Lie algebra is known. For example, the real semisimple Lie algebras were classified by Cartan, and so the classification of ...