Simple Lie algebras

Our task is then to classify the simply connected compact simple Lie groups. Fortunately, the complex simple Lie algebras have been completely classified. Unfortunately, a complex Lie algebra \({\mathfrak{g}}\) has several distinct real forms, i.e. real Lie algebras whose complexifications are \({\mathfrak{g}}\). Fortunately, there is always a unique (up to isomorphism) compact real form of \({\mathfrak{g}}\), which is the only one that corresponds to a compact simple Lie group. It turns out that the universal covering group of any compact connected semisimple group is compact, so that in particular, each complex simple Lie algebra corresponds to a unique real simply connected compact simple Lie group.

63.simple-lie-algebras

The complex simple Lie algebras consist of five exceptional Lie algebras \({g_{2}}\), \({f_{4}}\), \({e_{6}}\), \({e_{7}}\), and \({e_{8}}\), and four infinite series as follows.

Series Complex algebra Compact real form Other real form example
\({a_{n}\left(n\geq1\right)}\) \({sl\left(n+1,\mathbb{C}\right)}\) \({su\left(n+1\right)}\) \({su\left(n+1-s,s\right)}\)
\({b_{n}\left(n\geq2\right)}\) \({so\left(2n+1,\mathbb{C}\right)}\) \({so\left(2n+1\right)}\) \({so\left(2n+1-s,s\right)}\)
\({c_{n}\left(n\geq3\right)}\) \({sp\left(2n,\mathbb{C}\right)}\) \({sp\left(n\right)}\) \({sp\left(2n,\mathbb{R}\right)}\)
\({d_{n}\left(n\geq4\right)}\) \({so\left(2n,\mathbb{C}\right)}\) \({so\left(2n\right)}\) \({so\left(2n-s,s\right)}\)

Notes: The last column gives one example of a non-compact real form; others exist as well. \({sp(n)}\) complexifies to \({sp(2n,\mathbb{C})}\), which also has the real form \({sp(2n,\mathbb{R})}\).

The series start at the indicated value of \({n}\) to avoid duplicates in the list due to the following isomorphisms:

  • \({a_{1}\cong b_{1}\cong c_{1}}\): for example \({su(2)\cong so(3)\cong sp(1)}\)
  • \({b_{2}\cong c_{2}}\): for example \({so(5)\cong sp(2)}\)
  • \({d_{2}\cong a_{1}\oplus a_{1}}\): for example \({so(4)\cong so(3)\oplus so(3)}\)
  • \({d_{3}\cong a_{3}}\): for example \({so(6)\cong su(4)}\)

The third relation also leads to an isomorphism important in physics, \({so(3,1)\cong sl(2,\mathbb{C})}\). It is interesting to note that the series of simple Lie algebras roughly correspond to “rotations” in \({\mathbb{R}^{n}}\), \({\mathbb{C}^{n}}\), and \({\mathbb{H}^{n}}\); in fact, it turns out this idea can be extended to the exceptional Lie algebras, which can be related to transformations on \({\mathbb{O}}\). The derivation of this classification uses geometric objects in \({\mathbb{R}^{n}}\) that we will not discuss called root systems and associated combinatorial objects called Dynkin diagrams.

Completing our task, the compact real forms determine the simply connected compact simple Lie groups to be \({SU(n)}\) (for \({n\geq2}\)), \({Sp(n}\)) (for \({n\geq3}\)), \({\textrm{Spin}(n)}\) (for \({n=5}\) or \({n\geq7}\)), and the simply connected compact exceptional Lie groups \({G_{2}}\), \({F_{4}}\), \({E_{6}}\), \({E_{7}}\), and \({E_{8}}\). Here \({\textrm{Spin}(n)}\) is the universal covering group of \({SO(n)}\) for \({n>2}\), to be discussed in the section on rotations in terms of Clifford groups. Thus our final classification is:

Every compact connected Lie group is of the form \({G/N}\), where \({G}\) is a direct product of \({U(1)}\) and the above simple Lie groups, and \({N}\) is a normal discrete subgroup of \({G}\).

Regarding other types of Lie groups, we can note that above we have also classified the simple complex Lie algebras and groups. One can also show that any connected Lie group is topologically the product of a compact Lie group and a Euclidean space \({\mathbb{R}^{n}}\). We will end our discussion here, but we note that general Lie group theory provides various other statements concerning classifications.

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