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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 ...
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.
A Lie group is a group that has a compatible structure of a smooth manifold. 3. A Lie algebra is a vector space g {\displaystyle {\mathfrak {g}}} over a field F {\displaystyle F} with a binary operation [·, ·] (called the Lie bracket or abbr. bracket ) , which satisfies the following conditions: ∀ a , b ∈ F , x , y , z ∈ g ...
Its Lie algebra is the subspace of quaternion vectors. Since the commutator ij − ji = 2k, the Lie bracket in this algebra is twice the cross product of ordinary vector analysis. Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra. Standard treatments of Lie theory often begin with the classical groups.
In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true.
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.
Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.)
Lattice (group) 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