Compact Lie algebra

Properties

• Compact Lie algebras are reductive; note that the analogous result is true for compact groups in general.
• The Lie algebra ${\displaystyle \mathfrak{g}}$ for the compact Lie group G admits an Ad(G)-invariant inner product,. Conversely, if ${\displaystyle \mathfrak{g}}$ admits an Ad-invariant inner product, then ${\displaystyle \mathfrak{g}}$ is the Lie algebra of some compact group. If ${\displaystyle \mathfrak{g}}$ is semisimple, this inner product can be taken to be the negative of the Killing form. Thus relative to this inner product, Ad(G) acts by orthogonal transformations (${\displaystyle \mathrm{SO}(\mathfrak{g})}$) and ${\displaystyle \mathrm{ad}\mathfrak{g}}$ acts by skew-symmetric matrices (${\displaystyle \mathfrak{s}\mathfrak{o}(\mathfrak{g})}$). It is possible to develop the theory of complex semisimple Lie algebras by viewing them as the complexifications of Lie algebras of compact groups; the existence of an Ad-invariant inner product on the compact real form greatly simplifies the development.

  This can be seen as a compact analog of Ado's theorem on the representability of Lie algebras: just as every finite-dimensional Lie algebra in characteristic 0 embeds in ${\displaystyle \mathfrak{g}\mathfrak{l},}$ every compact Lie algebra embeds in ${\displaystyle \mathfrak{s}\mathfrak{o}.}$

• The Satake diagram of a compact Lie algebra is the Dynkin diagram of the complex Lie algebra with all vertices blackened.
• Compact Lie algebras are opposite to split real Lie algebras among real forms, split Lie algebras being "as far as possible" from being compact.

Classification

The compact Lie algebras are classified and named according to the compact real forms of the complex semisimple Lie algebras. These are:

• ${\displaystyle A_n:}$ ${\displaystyle {\mathfrak{s}\mathfrak{u}}_{n+1},}$ corresponding to the special unitary group (properly, the compact form is PSU, the projective special unitary group);
• ${\displaystyle B_n:}$ ${\displaystyle {\mathfrak{s}\mathfrak{o}}_{2n+1},}$ corresponding to the special orthogonal group (or ${\displaystyle {\mathfrak{o}}_{2n+1},}$ corresponding to the orthogonal group);
• ${\displaystyle C_n:}$ ${\displaystyle {\mathfrak{s}\mathfrak{p}}_n,}$ corresponding to the compact symplectic group; sometimes written ${\displaystyle {\mathfrak{u}\mathfrak{s}\mathfrak{p}}_n,}$;
• ${\displaystyle D_n:}$ ${\displaystyle {\mathfrak{s}\mathfrak{o}}_{2n},}$ corresponding to the special orthogonal group (or ${\displaystyle {\mathfrak{o}}_{2n},}$ corresponding to the orthogonal group) (properly, the compact form is PSO, the projective special orthogonal group);
• Compact real forms of the exceptional Lie algebras ${\displaystyle E_6,E_7,E_8,F_4,G_2.}$

Isomorphisms

The classification is non-redundant if one takes ${\displaystyle n\ge 1}$ for ${\displaystyle A_n,}$ ${\displaystyle n\ge 2}$ for ${\displaystyle B_n,}$ ${\displaystyle n\ge 3}$ for ${\displaystyle C_n,}$ and ${\displaystyle n\ge 4}$ for ${\displaystyle D_n.}$ If one instead takes ${\displaystyle n\ge 0}$ or ${\displaystyle n\ge 1}$ one obtains certain exceptional isomorphisms.

For ${\displaystyle n=0,}$ ${\displaystyle A_0\cong B_0\cong C_0\cong D_0}$ is the trivial diagram, corresponding to the trivial group ${\displaystyle \mathrm{SU}(1)\cong \mathrm{SO}(1)\cong \mathrm{Sp}(0)\cong \mathrm{SO}(0).}$

For ${\displaystyle n=1,}$ the isomorphism ${\displaystyle {\mathfrak{s}\mathfrak{u}}_2\cong {\mathfrak{s}\mathfrak{o}}_3\cong {\mathfrak{s}\mathfrak{p}}_1}$ corresponds to the isomorphisms of diagrams ${\displaystyle A_1\cong B_1\cong C_1}$ and the corresponding isomorphisms of Lie groups ${\displaystyle \mathrm{SU}(2)\cong \mathrm{Spin}(3)\cong \mathrm{Sp}(1)}$ (the 3-sphere or unit quaternions).

For ${\displaystyle n=2,}$ the isomorphism ${\displaystyle {\mathfrak{s}\mathfrak{o}}_5\cong {\mathfrak{s}\mathfrak{p}}_2}$ corresponds to the isomorphisms of diagrams ${\displaystyle B_2\cong C_2,}$ and the corresponding isomorphism of Lie groups ${\displaystyle \mathrm{Sp}(2)\cong \mathrm{Spin}(5).}$

For ${\displaystyle n=3,}$ the isomorphism ${\displaystyle {\mathfrak{s}\mathfrak{u}}_4\cong {\mathfrak{s}\mathfrak{o}}_6}$ corresponds to the isomorphisms of diagrams ${\displaystyle A_3\cong D_3,}$ and the corresponding isomorphism of Lie groups ${\displaystyle \mathrm{SU}(4)\cong \mathrm{Spin}(6).}$

If one considers ${\displaystyle E_4}$ and ${\displaystyle E_5}$ as diagrams, these are isomorphic to ${\displaystyle A_4}$ and ${\displaystyle D_5,}$ respectively, with corresponding isomorphisms of Lie algebras.