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I'm reading a book on integrable systems and am trying to understand Lie groups.

The author states a property I cannot understand:

Let me define the protagonists: L is the Lie derivative, m is an element of a Lie group and X and Y are in its Lie algebra. The dot, unless I'm mistaken, refers to a certain action of X on m.

I can understand the first and the last equality, the problem lies in the second equality. Why is it so? Is it true?

Trevor Gunn
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StarBucK
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1 Answers1

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Well, this comes from probably an older definition of a Lie Algebra, where someone defines for a given Lie Group $G$ its associated Lie Algebra, to be given by $$ \mathfrak{g}= \{ A \in M_{n \times n}(\mathbb{C}) \thinspace | \thinspace \forall t \in \mathbb{R}: \exp(tA) \in G\}. $$

So, if you have an action of the Lie group $G$ on a set $V$ (or vector space, module etc.), then this action defines a new one on the same $V$ of the the Lie Algebra this time, given as follows $$ \begin{split} \mathfrak{g} \times V &\rightarrow V, \\ (X, u) &\mapsto X.u= \frac{d}{dt}(\exp(tX).u)_{t=0}. \end{split} $$ You can check on your own that this is indeed a new action. What you see in the book is essentially this thing, applied into the corresponding Lie Bracket.

Somewhere in here: http://www.maths.gla.ac.uk/~ajb/dvi-ps/lie-bern.pdf you can find all the possible questions you can come up with. However, if you need something else please do let me know.

Tera
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  • Thank you very much! But the author is not talking about matrices at all. – StarBucK Jun 02 '17 at 22:35
  • If the Lie group is finite dimensional, then they truly do in fact. Check Ado's theorem. You haven't given basic information regarding your question, hence I thought that the above will be enough to help you out. However, even if you're talking about vector fields in your question or something else, the idea should be the same clearly. Try to work through what I've written out, in terms of your definitions and you will get your answer I believe. –  Jun 02 '17 at 23:27
  • Thank you for your quick and clear answer. Now it's time to work! Regards, Guilhem – StarBucK Jun 05 '17 at 10:04