Simple Lie Groups

\(\newcommand{\rar}{\rightarrow} \newcommand{\mb}{\mathbb} \newcommand{\mf}{\mathfrak}\)Here we'll establish some general framework around the notion of being simple.

Recall that we mentioned that given a group \(G\) and a subgroup \(N\) invariant under the adjoint action of \(G\) on itself, usually called a normal subgroup, we can form a quotient group \(G/N\). Similarly, for a Lie algebra \(\mf g\), an ideal \(I\) is a subspace invariant under the adjoint action \(ad(X)(Y) = [X,Y]\), and given an ideal we can form a quotient Lie algebra \(\mf g/I\).
We can bind these two ideas together by going to Hopf algebras. For a Hopf algebra \(H\), we write $$\Delta(x) = \sum_i x_{(1i)}\otimes x_{(2i)}$$ mimicking Sweedler notation, and define the adjoint action of \(H\) on itself as $$Ad(x)y =  \sum_i x_{(1i)}yS(x_{(2i)})$$ which again preserves the Hopf algebra structure if \(H\) is cocommutative. You can see that this action gives the group adjoint action when \(H = \mb kG\), and the Lie algebra adjoint action when \(H = U(\mf g)\).
When there are no nontrivial invariant subthings under the adjoint action, i.e. no quotient groups or quotient Lie algebras, we say that we have a simple group or a simple Lie algebra. Except for the 1-dimensional Lie algebra, which is not considered simple most of the time. It's too trivial, in that the bracket vanishes.

For Lie groups, the notion of being simple is tied to the Lie algebra: a simple Lie group is a Lie group whose Lie algebra is simple. This does not mean that a simple Lie group is simple as a group; it can still have nontrivial normal subgroups. However, because the Lie algebra is simple, the normal subgroups have to be \(0\)-dimensional as Lie groups, i.e. isolated points.
Most interesting properties of simple Lie groups can be determined by the Lie algebra, which is why we're not so concerned about those pesky normal subgroups. For "connected" simple Lie groups, which covers all the cases we'll be considering, those normal subgroups are constrained to being in the center, i.e. in the set of elements that commute with everything. The center of a group is always a normal subgroup, and for the connected simple Lie groups the center is a finite set of points.

The finite simple groups and the finite-dimensional simple Lie algebras over \(\mb R\) and \(\mb C\) have been classified, and in both cases the setup is a few infinite families indexed by integers along with a finite set of outliers that don't fit any particular pattern. For the simple Lie algebras over \(\mb C\), we have the following four families: \(\mf a_n = sl_{n+1}\), \(\mf b_n = so_{2n+1}\), \(\mf c_n = sp_{2n}\) and \(\mf d_n = so_{2n}\), where \(n\) can be any positive integer. We also have five exceptional simple Lie algebras, \(\mf e_6, \mf e_7, \mf e_8, \mf f_4,\) and \(\mf g_2\).
The great thing about the simple Lie algebras over \(\mb C\) is that all of their finite dimensional representations are fully decomposable. In other words, any finite dimensional representation can be written uniquely as a direct sum of irreducible representations. Contrast this to the case of, say, the Lie algebra of upper triangular matrices, where the obvious representation is not decomposable but is also not irreducible. Moreover, the irreducible representations of a finite-dimensional simple Lie algebra are easy to describe just from a few bits of data about the Lie algebra itself.
Over the reals, we have a somewhat more complicated picture for both the classification of the Lie algebras themselves and the representations. For the Lie algebras there are several real forms of each object on the list above. For instance, both \(sl_2(\mb R)\) and \(so_3(\mb R)\) are real forms of \(\mf a_1\). For the representations there is a bit of trickiness in terms of how representations decompose, since we have to worry about whether certain eigenvalues are real or not.

The classification of the finite simple groups is much more complicated, but is tightly linked to the classification of simple Lie algebras as most of the finite simple groups are "groups of Lie type", meaning that we take the field \(\mb k\) to be a finite field, build a Lie group over that field, and then fiddle a bit to get rid of some extra bits.

Unfortunately, the adjoint action does not in general yield the proper notion of ideal or coideal for Hopf algebra quotients of Hopf algebras. For algebras in general, we have to go back to the notion of ideal which uses the left- and right- regular actions, i.e. multiplication rather than the Lie bracket or its analogues. But we can still define a simple algebra as one with no proper ideals, i.e. with no proper quotients.