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  2. Lie group - Wikipedia

    en.wikipedia.org/wiki/Lie_group

    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 ...

  3. Table of Lie groups - Wikipedia

    en.wikipedia.org/wiki/Table_of_Lie_groups

    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.

  4. Simple Lie group - Wikipedia

    en.wikipedia.org/wiki/Simple_Lie_group

    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.

  5. Lie group–Lie algebra correspondence - Wikipedia

    en.wikipedia.org/wiki/Lie_groupLie_algebra...

    In mathematics, Lie groupLie 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.

  6. Glossary of Lie groups and Lie algebras - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_Lie_groups_and...

    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 ...

  7. Lie theory - Wikipedia

    en.wikipedia.org/wiki/Lie_theory

    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.

  8. Lie group action - Wikipedia

    en.wikipedia.org/wiki/Lie_group_action

    Let :, (,) be a (left) group action of a Lie group on a smooth manifold ; it is called a Lie group action (or smooth action) if the map is differentiable. Equivalently, a Lie group action of G {\displaystyle G} on M {\displaystyle M} consists of a Lie group homomorphism G → D i f f ( M ) {\displaystyle G\to \mathrm {Diff} (M)} .

  9. F4 (mathematics) - Wikipedia

    en.wikipedia.org/wiki/F4_(mathematics)

    In mathematics, F 4 is a Lie group and also its Lie algebra f 4. It is one of the five exceptional simple Lie groups. F 4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.