Lately in Lie Groups

Over the past few weeks here are some randome things about lie groups

some notes: lie8-10ish

\mathfrak{g} is solvable iff for an ideal \mathfrak{h} it and \mathfrak{g}/\mathfrak{h} are both solvable.  In fact there is a unique maximal solvable ideal and

the radical Rad (\mathfrak{g}) = \mathfrak{g}/\mathfrak{m} where \mathfrak{m} is the unique maximal solvable ideal.

The next big thing that was covered was the Baker Cambell Hasudorff formula which basically expresses \log(e^Xe^Y) as a series whose terms are in the ideal generated by X,Y.  There are some complicated formulas about the derivative of the exponential map, this is section of 3.3 of Hall.

After that was the classification of lie algebras of low dimension.  To describe a lie algebra its enough to describe what the lie bracket does to some basis, so to classifiy a lie algebra a good tool to use is looking at the bracket as a linear map [,] \colon \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g} and look at the various possible dimension the image can have. Some of the details are done in the posted notes.

Then there was a discussion about connected lie subgroup (analytic subgroups).  They are characterized as subgroups H \subset G whose lie algebra \mathfrak{h} is a subspace of \mathfrak{g} and any h \in H can be written as a product h = e^{X_1}\cdots e^{X_k}.

The big result is that if G has lie algebra \mathfrak{g} then sub algebras of \mathfrak{g} correspond to connected lie subgroups of G.

Another big result is that if G is simply connected with lie algebra \mathfrak{g} and H has lie algebra \mathfrak{h} then a lie algebra homomorphism \mathfrak{g} \to \mathfrak{h} is enough to determine a lie group homomorphism G \to H that is compatible with the exponential map.


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