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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 ...
This article gives a table of some common Lie groups and their associated Lie algebras.. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).
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 ...
Length of a Weyl group element; Lie algebra; Lie algebra extension; Lie group action; Lie group decomposition; Lie group–Lie algebra correspondence; Lie groupoid; Lie point symmetry; Lie product formula; Lie–Palais theorem; Lie's third theorem; Linear flow on the torus; Lorentz group
In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups.They are essential technical tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated homogeneous spaces.
In these cases the Lie algebra parameters have names: angle, hyperbolic angle, and slope. [2] These species of angle are useful for providing polar decompositions which describe sub-algebras of 2 x 2 real matrices. [3] There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3 ...
Every semisimple Lie group can be formed by taking a product of simple Lie groups and quotienting by a subgroup of its center. In other words, every semisimple Lie group is a central product of simple Lie groups. The semisimple Lie groups are exactly the Lie groups whose Lie algebras are semisimple Lie algebras.
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 .