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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.
The generator of any continuous symmetry implied by Noether's theorem, the generators of a Lie group being a special case. In this case, a generator is sometimes called a charge or Noether charge, examples include: angular momentum as the generator of rotations, [3] linear momentum as the generator of translations, [3]
Associated with every Lie group is its Lie algebra, a linear space of the same dimension as the Lie group, closed under a bilinear alternating product called the Lie bracket. The Lie algebra of SO(3) is denoted by s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} and consists of all skew-symmetric 3 × 3 matrices. [ 7 ]
The applications of Lie groups to differential systems were mainly established by Lie and Emmy Noether, and then advocated by Élie Cartan. Roughly speaking, a Lie point symmetry of a system is a local group of transformations that maps every solution of the system to another solution of the same system.
In mathematics, the term infinitesimal generator may refer to: an element of the Lie algebra, associated to a Lie group; Infinitesimal generator (stochastic processes), of a stochastic process; infinitesimal generator matrix, of a continuous time Markov chain, a class of stochastic processes; Infinitesimal generator of a strongly continuous ...
The special unitary group SU(n) is a strictly real Lie group (vs. a more general complex Lie group). Its dimension as a real manifold is n 2 − 1. Topologically, it is compact and simply connected. [2] Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). [3]
This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are locally isomorphic, and more strongly, they have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) have isomorphic Lie algebras, but SU(2) is a simply connected double cover of SO(3).