16 Apr 2014 No Comments

## Chapter 3: The Lie Algebra of the Lie Group and Canonical Co-ordinates

We now turn to the Lie algebra, which we shall define in the same way as [Rossmann] does in the first chapter. (Lie Algebra of a Connected Lie Group): For a connected Lie group $(\G,\,\bullet)$ as defined by Axioms 1 through 5, the Lie algebra $\g$ is the set of all tangent vectors at the […]

22 Apr 2014 No Comments

## Chapter 5: The Exponential Map

The aim of this post is to explore one parameter groups within a connected Lie group, namely those defined as follows: (One Parameter Subgroup): A one parameter subgroup of a connected Lie group is a homomorphic image $\mathfrak{P}\subset \G$ of the group $(\R,\,+)$ of reals with addition within $\G$. In other words, there is a […]

by Rod Vance in Lie Groups: What They Are, Lie Theory, Part 1: The Local View Tags: Cauchy Initial Value Problem, Differential Equation, Exponential Map, Flow, Flow Equation, Isomorphism, Lie Algebra, Lie Group, One Parameter Group, Peano Existence Theorem, Picard-Lindelöf Theorem