13 May 2014 No Comments

## Chapter 15: How Unique is the Lie Group Structure for a Given Lie Group?

We can think of our fundamental axioms 1 through 5 as a “specification” that we can lay down on an abstract group to beget a Lie group and ultimately, as we have shown, an analytic manifold structure. Can we do this in more than one way? How unique is the Lie group structure? Once we […]

18 Jun 2014 No Comments

## Chapter 16: The Killing Form: A Powerful Insight into Global Lie Group Topology and Abstract Group Structure

The Killing Form is named after Wilhelm Killing (1847 – 1923), thus explaining its otherwise startling (for English speakers) name. Riemannian geometry even has Killing Fields, i.e. vector fields whose flows conserve the metric for the Riemannian manifold. The Killing form is a homogeneous, billinear, symmetric binary real-valued form defined on a Lie algebra $\g$ […]

by Rod Vance in Lie Theory, Part 2: The Global View Tags: Associative Form, Invariant Form, Irreducible Representation, Killing Form, Representation, Schur's Lemma, Semisimple Lie Algebra