Relationships between Lie groups and Lie algebras

The exponential map is a diffeomorphism in some neighborhood of the identity, but in general over \({G}\) it is neither injective nor surjective. This reflects the fact that in general there are an infinite number of different Lie groups with the same Lie algebra. However, some facts regarding the relationship between a finite-dimensional Lie algebra and its corresponding Lie groups are:

  • The exponential map is surjective for any compact connected Lie group
  • Any connected Lie group is generated by a neighborhood of the identity, i.e. every element is a finite product of exponentials

The relationship between Lie groups and Lie algebras also extends to derived objects:

  • There is a one-to-one correspondence between the connected Lie subgroups of \({G}\) and the Lie subalgebras of \({\mathfrak{g}}\)
  • A connected Lie subgroup of a connected \({G}\) is normal iff its Lie algebra is an ideal in \({\mathfrak{g}}\)
  • The Lie algebra of \({G\times H}\) is \({\mathfrak{g}\oplus\mathfrak{h}}\)
  • Every Lie group homomorphism \({\phi\colon G\to H}\) determines a Lie algebra homomorphism \({\mathrm{d}\phi\colon\mathfrak{g}\to\mathfrak{h}}\); the converse holds if \({G}\) is simply connected

Lie algebras can be seen to restrict the topology of Lie groups as compared to general manifolds. For example, a basis of \({T_{e}G}\) corresponds to linearly independent left-invariant vector fields on all of \({G}\); therefore every Lie group is orientable and parallelizable. The only connected one-dimensional Lie groups are \({\mathbb{R}}\) and \({S^{1}}\) (under addition of value and angle). Both are abelian, and in fact any connected abelian Lie group is a direct product of these one-dimensional Lie groups. In particular, the only compact 2-dimensional Lie group is the torus \({T^{2}=S^{1}\times S^{1}}\).

An Illustrated Handbook