I am a physicist and we do Lie algebras pretty informally, so I hope my question makes any sense to a mathematician. There is one thing that I don't quite understand, which is why we can find a basis for the elements of the Lie algebra?

Just to show you at what kind of informal level I understand the concepts so far, here is my line of thought: We have a group and find a set of matrices that form a representation for the group. Then for every such matrix $O$ we find a corresponding matrix in the Lie algebra $H$ such that: $O = e^{iH}$. Now we say that there we can find a basis for these elements, i.e. a finite set $H_i$ such that for any $H$ exist $\alpha_j$: $H=\sum_j \alpha_j H_j$. If anything so far is incorrect please let me know.

Now regarding my actual question: I thought to be able decompose elements like this would that not require the property that for 3 elements $O_1, O_2, O_3$ with $O_3 = O_1 O_2$ we also have

\begin{equation} \alpha_j^{(3)} = \alpha_j^{(1)}+\alpha_j^{(2)} \end{equation}

(where $\alpha_j^{(1)}$ are the Lie algebra coefficients of $O_1$ and similarly for 2, 3)? This is of course not true since the generators do not commute. What am I missing? Is there maybe a different relation between the coefficients of 3 group elements that are related by a multiplication other than adding them?

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    $\begingroup$ I'm not sure I understand : the Lie algebra of a Lie group is in particular a vector space, so there is a basis. $\endgroup$ – Captain Lama Apr 11 '16 at 16:34
  • $\begingroup$ @CaptainLama I think my confusion is about the relation between this vector space and the group elements. I.e. I don't understand how come that the group element corresponding to the sum of two vectors in the Lie algebra does not correspond to the group product of the two elements corresponding to the two vectors? $\endgroup$ – Wolpertinger Apr 11 '16 at 16:43
  • $\begingroup$ Does this help: en.wikipedia.org/wiki/…? $\endgroup$ – Hans Lundmark Apr 11 '16 at 17:51
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    $\begingroup$ I think your confusion comes from assuming that $e^{x+y} = e^xe^y$ still holds when $x, y$ are operators. It does not hold in general. $\endgroup$ – Paul Sinclair Apr 11 '16 at 19:06
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    $\begingroup$ The Lie algebra is not defined by the exponential function. Instead it is the archetype tangent space for any point in the Lie group. In particular, you can consider it the tangent space at the identity. As such it is naturally a vector space. $\endgroup$ – Paul Sinclair Apr 12 '16 at 16:19

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