3 - From Lie group to Lie algebra

Summary

In the previous chapter, we discussed the tangent structure on a Lie group, and the relationship between the set of one parameter subgroups and the tangent space T_eG at the identity element. The most striking feature of the tangent space at the identity of a Lie group is the existence of a natural product, called a Lie bracket, so that T_eG is an algebra; the Lie algebra of the Lie group.

Since Lie groups arise in different formulations, so does the appearance of the bracket in the Lie algebra. However, they all follow from the one formula for the Lie bracket of two vector fields on ℝⁿ which we consider in the first section. The geometric formulation looks unusable in practice, so we ‘deconstruct’ it to make it easily computable, prove some of its properties and discuss the all important Frobenius Theorem. We then derive the Lie algebra bracket for a general Lie group in Section 3.2, giving details in the two main cases of interest, matrix groups in Section 3.2.1 and transformation groups in Section 3.2.2. Although many authors simply give the formulae for the Lie bracket in these two cases as the definition of the Lie bracket, and readers only needing to compute can skip straight to these formulae, it is both interesting and helpful to know that in fact they are both instances of the same geometric construction.