Search results
Results from the WOW.Com Content Network
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 ...
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.
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.
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 ...
A connected complex Lie group that is a compact group is abelian and a connected compact complex Lie group is a complex torus; i.e., a quotient of by a lattice. Let A be a compact abelian Lie group with the identity component .
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.
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
1. A simple Lie group is a connected Lie group that is not abelian which does not have nontrivial connected normal subgroups. 2. A simple Lie algebra is a Lie algebra that is non abelian and has only two ideals, itself and {}. 3.