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In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces.
A finite-dimensional simple complex Lie algebra is isomorphic to either of the following: , , (classical Lie algebras) or one of the five exceptional Lie algebras. [1]To each finite-dimensional complex semisimple Lie algebra, there exists a corresponding diagram (called the Dynkin diagram) where the nodes denote the simple roots, the nodes are jointed (or not jointed) by a number of lines ...
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).
Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras. [30] They are central extensions of products of simple Lie groups. The identity component of any Lie group is an open normal subgroup , and the quotient group is a discrete group .
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.
Sp(2n, R) is a real, non-compact, connected, simple Lie group. [4] It has a fundamental group isomorphic to the group of integers under addition. As the real form of a simple Lie group its Lie algebra is a splittable Lie algebra. Some further properties of Sp(2n, R): The exponential map from the Lie algebra sp(2n, R) to the group Sp(2n, R) is ...
These groups (the groups of Lie type, together with the cyclic groups, alternating groups, and the five exceptional Mathieu groups) were believed to be a complete list, but after a lull of almost a century since the work of Mathieu, in 1964 the first Janko group was discovered, and the remaining 20 sporadic groups were discovered or conjectured ...
The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the representation theory of Lie groups. For finite-dimensional representations, there is an equivalence of categories between representations of a real Lie algebra and representations of the corresponding simply ...