Definition of Lie Group and its Algebras

[Greg Bernhardt]

Definition/Summary

A Lie group ("Lee") is a continuous group whose group operation on its parameters is differentiable in them.

Lie groups appear in a variety of contexts, like space-time and gauge symmetries, and in solutions of certain differential equations.

The elements of Lie groups can be determined from elements that are close to the identity. The differences between those elements and the identity element form the tangent space of the identity, and that tangent space's algebra is the group's Lie algebra.

Lie algebras are often much easier to study than the groups that they are derived from, and many properties of those groups can be determined from the properties of those algebras. However, groups with isomorphic Lie algebras may nevertheless be nonisomorphic. A classic case is the rotation group SO(n) and the spinor group Spin(n). Though their algebras are isomorphic, Spin(n) is the double cover of SO(n), and elements x and -x of Spin(n) map onto the same element of SO(n).

Equations

Elements of a Lie group for parameter set x can be found from elements of the group's Lie algebra L using the exponential map:
$D(x) = e^{x \cdot L} D(0)$

For a matrix group,
$e^X = \sum_{n=0}^{\infty} \frac{X^n}{n!} = I + \frac{1}{1!}X + \frac{1}{2!}X^2 + \frac{1}{3!}X^3 + \cdots$

using the exponential function's Taylor series.

Extended explanation

Here are some Lie groups:

• The general linear group GL(n,X) of invertible n-dimensional matrices of elements of continuous field X. For GL(n,R), one must use those matrices with positive determinants; GL(n,C) does not have that restriction.
• The special linear group SL(n,X), a subgroup of GL(n,X) whose elements have determinant 1. GL(n,X) = SL(n,X) * GL(1,X)
• The group of n-dimensional complex unitary matrices U(n).
• The group of special ones SU(n), a subgroup of U(n) whose elements have determinant 1. U(n) = SU(n) * U(1)
• The group of n-dimensional real orthogonal matrices with determinant 1, SO(n).
• However, the group without that determinant restriction, O(n), is not quite a Lie group, because its element determinants are either 1 (pure rotations) or -1 (rotation-reflections). For odd n, O(n) = SO(n) * {I,-I}, while for even n, -I is in SO(n). But in both cases, O(n) / SO(n) = Z(2).
• The group of spinors related to SO(n): Spin(n)
• The symplectic or quaternionic group Sp(2n), whose elements are real matrices D that satisfy D.J.D^T = J, where J is a symplectic form. It is antisymmetric, and J.J is proportional to the identity matrix. A convenient form of it is
  $J = \begin{pmatrix} 0 & -I_n \\ I_n & 0 \end{pmatrix}$

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[fresh_42]

See also:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
and the sources at the end of the article.

A Lie group is a topological group with a compatible analytic structure, i.e. an analytic manifold with analytic group operations. [Varadarajan]

Analytic means locally a convergent power series, which implies smoothness.