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Good evening !

I am studying mathematical Gauge theories from the book from Mark J.D. Hamilton's book on mathematical gauge theory.

The introduction in this book is about Lie groups and Lie algebras. I got stuck on the following example :

"Every countable group $G$ with the manifold structure as a discrete space, i.e. a countable union of isolated points, is a $0$-dimensional Lie group, because every map $G\times G \rightarrow G$ is smooth (locally constant)."

I know what a countable group is, and I think I do understand the "with the manifold structure as a discrete space" part. ( I imagine it as a mesh of points (g,h)/ g,h $\in$ $G$ and at each point we have $g+h$ / $+$ is the group operation) However, I am having a hard time understanding the smoothness of such a structure (which I imagine as being a discrete 2D surface in a 3D space). I would appreciate it if someone can better clarify the example to me. (Or recommend a book that is useful for self-studying Lie groups and Lie algebras) P.S. I am a physics student (this is why I didn't use a lot of technical terms), however I am highly interested in mathematics and I might continue later on in the domain of mathematical physics !

Thanks in advance

Edit : I would also appreciate it if someone could give me an example of a $0$-dimensional Lie group

Ceethemez
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    Do you know what a smooth manifold is? – Michael Albanese Jul 09 '21 at 15:08
  • No, other than the definition I just read (and understood on google ), I don't know what a smooth manifold is. – Ceethemez Jul 10 '21 at 08:22
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    Well, a Lie group is a smooth manifold with additional structure, so I recommend you learn about smooth manifolds first. Moreover, if you want to learn gauge theory, you need to have a solid understanding of smooth manifolds. There are many excellent books for this, my suggestion is Lee's Introduction to Smooth Manifolds, but there are many others, see this question. – Michael Albanese Jul 10 '21 at 13:01
  • To understand smooth manifolds do I need some knowledge in differential geometry (and some topology) ? – Ceethemez Jul 10 '21 at 22:49
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    Yes. Differential geometry is the study of smooth manifolds. – Michael Albanese Jul 11 '21 at 01:56

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