Lie Groups

A Lie group is a group G which is also an analytic manifold such that the mapping (a, b) → ab⁻¹ (a, b ∈ G) of the product manifold G × G into G is analytic. Thus a Lie group is set endowed with compatible structures of a group and an analytic manifold. A subgroup H of a Lie group G is said to be a Lie subgroup if it is a submanifold of the underlying manifold of G. A group G is called a Lie transformation group of a differentiable manifold M if there is a differentiable map φ : G × M → M, φ(g; x) = gx(x ∈ M) such that (i) (g₁ · g₂)x = g₁ · (g₂x) for x ∈ M and (ii) ex = x for the identity element e of G and x ∈ M…