In themathematical field ofLie theory, there aretwo definitions of acompactLie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of acompact Lie group;^[1] this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whoseKilling form isnegative definite; this definition is more restrictive and excludes tori.^[2] A compact Lie algebra can be seen as the smallestreal form of a corresponding complex Lie algebra, namely the complexification.

Formally, one may define a compact Lie algebra either as the Lie algebra of a compactLie group, or as a real Lie algebra whose Killing form is negative definite. These definitions do not quite agree:^[2]

• The Killing form on the Lie algebra of a compact Lie group isnegativesemidefinite, not negative definite in general.
• If the Killing form of a Lie algebra is negative definite, then the Lie algebra is the Lie algebra of a compactsemisimple Lie group.

In general, the Lie algebra of a compact Lie group decomposes as the Lie algebra direct sum of a commutative summand (for which the corresponding subgroup is a torus) and a summand on which the Killing form is negative definite.

It is important to note that the converse of the first result above is false: Even if the Killing form of a Lie algebra is negative semidefinite, this does not mean that the Lie algebra is the Lie algebra of some compact group. For example, the Killing form on the Lie algebra of theHeisenberg group is identically zero, hence negative semidefinite, but this Lie algebra is not the Lie algebra of any compact group.

• Compact Lie algebras arereductive;^[3] note that the analogous result is true for compact groups in general.
• The Lie algebra${\displaystyle \mathfrak{g}}$ for the compact Lie groupG admits an Ad(G)-invariantinner product,.^[4] Conversely, if${\displaystyle \mathfrak{g}}$ admits an Ad-invariant inner product, then${\displaystyle \mathfrak{g}}$ is the Lie algebra of some compact group.^[5] 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 byorthogonal transformations (${\displaystyle \mathrm{SO}(\mathfrak{g})}$) and${\displaystyle \mathrm{ad}\mathfrak{g}}$ acts byskew-symmetric matrices (${\displaystyle \mathfrak{s}\mathfrak{o}(\mathfrak{g})}$).^[4] It is possible to develop the theory of complex semisimple Lie algebras by viewing them as the complexifications of Lie algebras of compact groups;^[6] 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 ofAdo'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${\displaystyle \mathfrak{s}\mathfrak{o}.}$

• TheSatake diagram of a compact Lie algebra is theDynkin diagram of the complex Lie algebra withall vertices blackened.
• Compact Lie algebras are opposite tosplit real Lie algebras amongreal forms, split Lie algebras being "as far as possible" from being compact.

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

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

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 certainexceptional 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 orunit 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.

• Real form
• Split Lie algebra

• Lie group, compact, inEncyclopaedia of Mathematics

• Properties of Lie algebras

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