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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. Together with the commutative Lie group of the real numbers, , and that of the unit-magnitude ...
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 .
The symmetry group of the sphere (n =3) or hypersphere. SO (1) is a single point and SO (2) is isomorphic to the circle group, SO (3) is the rotation group of the sphere. special euclidean group: group of rigid body motions in n-dimensional space. For n =1: isomorphic to S 1.
e. In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of simple Lie algebras is called a semisimple Lie algebra. A simple Lie group is a connected Lie group ...
orthogonal groups. Chevalley groups, Cn (q) n > 2. symplectic groups. Chevalley groups, Dn (q) n > 3. orthogonal groups. Simplicity. A1 (2) and A1 (3) are solvable, the others are simple. B2 (2) is not simple but its derived group B2 (2)′ is a simple subgroup of index 2; the others are simple. All simple.
The simple Lie algebras are classified by the connected Dynkin diagrams. Every semisimple Lie algebra over an algebraically closed field of characteristic 0 is a direct sum of simple Lie algebras (by definition), and the finite-dimensional simple Lie algebras fall in four families – A n, B n, C n, and D n – with five exceptions E 6, E 7, E ...
In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic. (The Tits group is sometimes regarded as a sporadic group ...
The goal of E 8 Theory is to describe all elementary particles and their interactions, including gravitation, as quantum excitations of a single Lie group geometry—specifically, excitations of the noncompact quaternionic real form of the largest simple exceptional Lie group, E 8. A Lie group, such as a one-dimensional circle, may be ...