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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 ...
See Table of Lie groups for a list. General linear group, special linear group. SL 2 (R) SL 2 (C) Unitary group, special unitary group. SU(2) SU(3) Orthogonal group, special orthogonal group. Rotation group SO(3) SO(8) Generalized orthogonal group, generalized special orthogonal group. The special unitary group SU(1,1) is the unit sphere in the ...
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry.
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.
In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups.They are essential technical tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated homogeneous spaces.
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 representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra. [1] [2] There is a closely related theorem classifying the irreducible representations of a connected compact Lie group. [3]
Since is a Lie group automorphism, Ad g is a Lie algebra automorphism; i.e., an invertible linear transformation of to itself that preserves the Lie bracket. Moreover, since g ↦ Ψ g {\displaystyle g\mapsto \Psi _{g}} is a group homomorphism, g ↦ Ad g {\displaystyle g\mapsto \operatorname {Ad} _{g}} too is a group homomorphism. [ 1 ]