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Mathematics for Physics

176 Lie

A set of matrices with a mapping for products and a mapping for inverses that is used to describe continuous change of vectors by linear transformations.

Note: Also called Lie group.

definition [d] (Lie = Lie Group) From Hassani Definition 29.1.1: a Lie group \(G\) is a differentiable manifold endowed with a group structure such that the group operation \(G \times G \rightarrow G\) and the map \(G \rightarrow G\) given by \(g \mapsto g^{-1}\) are differentiable.

definition [d] (Lie = Lie Group) From Tu Definition 15.1: a Lie group is a \(C^{\infty}\) manifold \(G\) that is also a group such that the multiplication map

  • \(\mu: G \times G \rightarrow G\)

and the inverse map

  • \(\iota: G \rightarrow G\), \(\iota(x) = x^{-1}\)

are both \(C^{\infty}\).

where

  • \(G\) is the Lie group.
  • \(\mu\) is the multiplication map.
  • \(\iota\) is the inverse map.
  • \(C^{\infty}\) means infinitely differentiable.

176.1 Elementary Example

176.1.1 Simple

A discrete group closed under multiplication and inverses illustrates the group maps \(\mu\) and \(\iota\).

\[ G = \{ e,\ a,\ a^{2} \} \]

\[ \mu(a,a) = a^{2},\quad \mu(a,a^{2}) = e,\quad \iota(a) = a^{2} \]

where

  • \(G\) is the cyclic group of order three.
  • \(e\) is the identity element.
  • \(a\) is a generator with \(a^{3} = e\).
  • \(\mu : G \times G \rightarrow G\) is multiplication.
  • \(\iota : G \rightarrow G\) is inversion.

176.1.2 General

The fourth roots of unity form a four-element group under complex multiplication, with the same maps \(\mu\) and \(\iota\).

\[ G = \{ 1,\ i,\ -1,\ -i \} \]

\[ \mu(i,i) = -1,\quad \iota(i) = -i,\quad \mu(-i,-i) = -1 \]

where

  • \(i\) is the imaginary unit with \(i^{2} = -1\).
  • \(\mu\) and \(\iota\) are the group multiplication and inverse maps.