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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.
Then a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly ...
A semisimple Lie group is a connected Lie group so that its only closed connected abelian normal subgroup is the trivial subgroup. Every simple Lie group is semisimple. More generally, any product of simple Lie groups is semisimple, and any quotient of a semisimple Lie group by a closed subgroup is semisimple.
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.
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 ...
Pages in category "Lie groups" The following 164 pages are in this category, out of 164 total. ... Table of Lie groups; Tangent Lie group; Theory of Lie groups; Theta ...
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 .
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way , (,) is holomorphic. Basic examples are GL n ( C ) {\displaystyle \operatorname {GL} _{n}(\mathbb {C} )} , the general linear groups over the complex numbers .