understanding relevance of Lie vs topological groups

Question

A silly easy to state question. When dealing with topological groups, I'm trying to understand more profoundly the advantages of having a Lie group structure against just a topological one. Can somebody please illustrate this more in depth?

I understand what it implies right away, say having topological against differentiable manifolds and so on, but I guess I could really use examples/good guidance in this specific case to grasp better the full implications of being a Lie group?

sorry if i'm being too vague here, any help is happily appreciated.

Answer 1

Naturally, Lie groups admit enough structure to support differential calculus and associated tools.

Conversely, being a topological group (rather than a Lie group) is not analogous to being a topological manifold (rather than a smooth manifold): "Most" topological groups aren't manifolds in any sense. For starters, think of the additive group of rationals or $p$-adic numbers, or the countable product of cyclic groups of order two with the product topology (whose total space is homeomorphic to the Cantor ternary set), or an irrational winding on a torus.

Answer 2

One of the most important features of Lie groups is that they can be studied by means of their Lie algebra. Difficult questions about the global geometry of a Lie group sometime translate to simpler algebraic questions about its Lie algebra.

For example, compact connected Lie groups can be classified in terms of Dynkin diagrams, which is an algebraic object associated to the Lie algebra.

A topological group has a priori no Lie algebra.