Compact Lie algebra

In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori, though allowing negative semidefinite includes tori and agrees with the previous definition. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification.

Definition

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

• The Killing form on the Lie algebra of a compact Lie group is negative semidefinite, 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 compact Lie group.

The difference is precisely in whether to include tori (and their corresponding Lie algebra, which is abelian and hence has trivial Killing form) or not: real Lie algebras with negative definite Killing forms correspond to compact semisimple Lie groups, while real Lie algebras with negative semidefinite Killing forms correspond to products of compact semisimple Lie groups and tori. One can distinguish between these by calling a Lie algebra with negative semidefinite Killing form a compact reductive Lie algebra, and a Lie algebra with negative definite Killing form a compact semisimple Lie algebra, which corresponds to reductive Lie algebras being direct sums of semisimple and abelian.

Properties

• Compact Lie algebras are reductive; note that the analogous result is true for compact groups in general.
• A compact Lie algebra $\mathfrak{g}$ for the compact Lie group G admits an Ad(G)-invariant inner product, and this property characterizes compact Lie algebras. This inner product can be taken to be the negative of the Killing form, and this is the unique Ad(G)-invariant inner product up to scale. Thus relative to this inner product, Ad(G) acts by orthogonal transformations ($\operatorname{SO}(\mathfrak{g})$) and $\operatorname{ad}\ \mathfrak{g}$ acts by skew-symmetric matrices ($\mathfrak{so}(\mathfrak{g})$).

  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 $\mathfrak{gl},$ every compact Lie algebra embeds in $\mathfrak{so}.$

• 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:

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

Isomorphisms

The exceptional isomorphisms of connected Dynkin diagrams yield exceptional isomorphisms of compact Lie algebras and corresponding Lie groups.

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

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

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

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

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

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

See also

• Real form
• Split Lie algebra

References

• .

External links

• Lie group, compact, V.L. Popov, in Encyclopaedia of Mathematics, ISBN 1-4020-0609-8, SpringerLink

• (Knapp 2002, Section 4, pp. 248–251)
• (Knapp 2002, Propositions 4.26, 4.27, pp. 249–250)
• (Knapp 2002, Proposition 4.25, pp. 249)
• (Knapp 2002, Proposition 4.24, pp. 249)
• SpringerLink