Compact real form of a complex semisimple Lie algebra

Intuition [Beginner]

Inside every complex semisimple Lie algebra there is a "compact" real version --- a real Lie algebra where the Killing form is negative definite. Think of the skew-hermitian matrices (matrices $A$ with $A^{\ast }=-A$, where $A^{\ast }$ is the conjugate-transpose) sitting inside all complex matrices. The skew-hermitian matrices form a real Lie algebra (not complex: multiplying by $i$ takes you out), and on this algebra the Killing form is always negative. This is the prototypical compact real form.

The word "compact" refers to the associated Lie group. The Lie algebra of skew-hermitian matrices exponentiates to the unitary group, which is compact (it fits inside a bounded region of matrix space). More generally, every compact real form exponentiates to a compact Lie group. The correspondence between compact Lie groups and complex semisimple Lie algebras is one of the deepest bridges in mathematics.

Why does this concept exist? The compact real form provides the analytic footing for the representation theory of complex semisimple Lie algebras. Weyl's unitary trick uses the compact real form to prove complete reducibility of representations by averaging over the compact group, transferring topological arguments from the compact side to the algebraic side.

Visual [Beginner]

A complex semisimple Lie algebra drawn as a large region, with a distinguished real subspace inside it. The real subspace is the compact real form, labelled with the Killing form being negative definite. An involution (a self-map that squares to the identity) acts on the whole algebra, and its fixed points are precisely the compact real form.

The involution splits the algebra into two pieces: the compact piece (fixed by the involution, where the Killing form is negative) and the non-compact piece (negated by the involution, where the Killing form is positive).

Worked example [Beginner]

The Lie algebra $\mathfrak{sl}(2,\mathbb{C})$ consists of all $2\times 2$ complex matrices with trace zero. It is 3-dimensional over $\mathbb{C}$ (6-dimensional over $\mathbb{R}$).

Step 1. A standard basis for $\mathfrak{sl}(2,\mathbb{C})$ is $H=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, $E=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, $F=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$, with bracket relations $[H,E]=2E$, $[H,F]=-2F$, $[E,F]=H$.

Step 2. The compact real form $\mathfrak{su}(2)$ consists of skew-hermitian trace-zero matrices: $2\times 2$ complex matrices $A$ with $A^{\ast }=-A$ and trace 0. A real basis is:

$$iH=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),E-F=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),i(E+F)=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right).$$

Step 3. The Killing form on $\mathfrak{su}(2)$ is $\kappa (X,Y)=4\mathrm{tr}(XY)$ for $2\times 2$ matrices. On the basis $iH,E-F,i(E+F)$, each has $\kappa$-norm $-8$: for instance, $\kappa (iH,iH)=4\mathrm{tr}((iH)(iH))=4\cdot (-2)=-8$. The Killing form is negative definite.

Step 4. The Lie group associated to $\mathfrak{su}(2)$ is $\mathrm{SU}(2)$, the group of $2\times 2$ unitary matrices with determinant 1. This group is diffeomorphic to the 3-sphere $S^3$ and is compact.

What this tells us: the 3-dimensional complex algebra $\mathfrak{sl}(2,\mathbb{C})$ contains a 3-dimensional real algebra $\mathfrak{su}(2)$ with negative-definite Killing form, exponentiating to the compact group $\mathrm{SU}(2)\cong S^3$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. A real form of $\mathfrak{g}$ is a real Lie subalgebra ${\mathfrak{g}}_0\subset \mathfrak{g}$ such that $\mathfrak{g}={\mathfrak{g}}_0\oplus i{\mathfrak{g}}_0$ (direct sum of real vector spaces). Equivalently, the canonical map $\mathbb{C}{\otimes }_{\mathbb{R}}{\mathfrak{g}}_0\to \mathfrak{g}$ is an isomorphism of complex Lie algebras.

A compact real form of $\mathfrak{g}$ is a real form $\mathfrak{u}$ such that the Killing form ${\kappa }_{\mathfrak{u}}$ (the restriction of the Killing form of $\mathfrak{g}$ to $\mathfrak{u}$) is negative definite: ${\kappa }_{\mathfrak{u}}(X,X)<0$ for all nonzero $X\in \mathfrak{u}$.

Definition (Cartan involution). Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{u}$ a compact real form. The Cartan involution associated to $\mathfrak{u}$ is the conjugation $\theta :\mathfrak{g}\to \mathfrak{g}$ defined by $\theta (X+iY)=X-iY$ for $X,Y\in \mathfrak{u}$. This is an involutive automorphism (${\theta }^2=\mathrm{id}$) whose fixed-point set is $\mathfrak{u}$.

Counterexamples to common slips

• A compact real form is not a complex subalgebra. If $X\in \mathfrak{u}$, then $iX\notin \mathfrak{u}$ in general (the Killing form of $iX$ is positive). So $\mathfrak{u}$ is only a real Lie algebra, not closed under multiplication by $i$.

• The Killing form on the whole complex algebra is not negative definite. On $\mathfrak{sl}(2,\mathbb{C})$, the Killing form $\kappa (H,H)=8>0$. Only on the compact real form is it negative definite.

• Not every real form is compact. For example, $\mathfrak{sl}(2,\mathbb{R})$ is a non-compact real form of $\mathfrak{sl}(2,\mathbb{C})$.

Key theorem with proof [Intermediate+]

Theorem (Existence of compact real forms). Every complex semisimple Lie algebra $\mathfrak{g}$ has a compact real form.

Proof. The proof constructs the compact real form explicitly from the root space decomposition. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and let $\Phi$ be the root system of $(\mathfrak{g},\mathfrak{h})$ with root spaces ${\mathfrak{g}}_{\alpha }$ for $\alpha \in \Phi$.

Step 1: Chevalley basis. For each $\alpha \in \Phi$, choose a root vector $e_{\alpha }\in {\mathfrak{g}}_{\alpha }$ so that the collection $\{h_i\}\cup \{e_{\alpha }{\}}_{\alpha \in \Phi }$ forms a Chevalley basis: the $h_i$ span $\mathfrak{h}$ and are derived from the simple coroots, and the structure constants are integers satisfying $[e_{\alpha },e_{-\alpha }]=h_{\alpha }$ (the coroot associated to $\alpha$) and $[e_{\alpha },e_{\beta }]=\pm (p+1)e_{\alpha +\beta }$ where $p$ is the largest integer with $\beta -p\alpha \in \Phi$.

Step 2: Define the real subalgebra. Let

$$\mathfrak{u}:={\mathrm{span}}_{\mathbb{R}}\{i{h}_1,\dots ,i{h}_r\}\cup \{e_{\alpha }-e_{-\alpha },i(e_{\alpha }+e_{-\alpha }):\alpha \in {\Phi }^{+}\}$$

where ${\Phi }^{+}$ is a choice of positive roots and $h_1,\dots ,h_r$ are the simple coroots.

Step 3: Verify $\mathfrak{u}$ is a real Lie algebra. The bracket of any two elements of $\mathfrak{u}$ lies in $\mathfrak{u}$. For the Chevalley basis elements:

$$[i{h}_k,e_{\alpha }-e_{-\alpha }]=i\alpha (h_k)(e_{\alpha }+e_{-\alpha })\in \mathfrak{u},$$

$$[e_{\alpha }-e_{-\alpha },i(e_{\alpha }+e_{-\alpha })]=i[e_{\alpha },e_{\alpha }]-i[e_{-\alpha },e_{-\alpha }]+i[e_{\alpha },e_{-\alpha }]-i[e_{-\alpha },e_{\alpha }]=2i{h}_{\alpha }\in \mathfrak{u}.$$

The remaining bracket computations follow similarly from the integrality of the Chevalley basis structure constants, which ensures closure over $\mathbb{R}$.

Step 4: Verify negative definiteness. The Killing form on the basis of $\mathfrak{u}$ satisfies:

• $\kappa (i{h}_k,i{h}_k)=-\kappa (h_k,h_k)<0$ (since $\kappa$ is positive definite on ${\mathfrak{h}}_{\mathbb{R}}={\mathrm{span}}_{\mathbb{R}}\{h_i\}$).
• $\kappa (e_{\alpha }-e_{-\alpha },e_{\alpha }-e_{-\alpha })=\kappa (e_{\alpha },e_{\alpha })+\kappa (e_{-\alpha },e_{-\alpha })=-2\kappa (e_{\alpha },e_{-\alpha })<0$ (using $\kappa (e_{\alpha },e_{\beta })=0$ for $\alpha +\beta \ne 0$ and $\kappa (e_{\alpha },e_{-\alpha })>0$).
• Similarly for $i(e_{\alpha }+e_{-\alpha })$.

Since the Killing form is negative on each basis element and the root spaces are orthogonal, $\kappa$ is negative definite on $\mathfrak{u}$.

Step 5: Verify $\mathfrak{u}$ is a real form. The dimension of $\mathfrak{u}$ over $\mathbb{R}$ equals $r+2|{\Phi }^{+}|={\dim }_{\mathbb{C}}\mathfrak{g}$, and $\mathfrak{g}=\mathfrak{u}+i\mathfrak{u}$ with $\mathfrak{u}\cap i\mathfrak{u}=0$. So $\mathfrak{g}=\mathfrak{u}\oplus i\mathfrak{u}$. $\square$

Bridge. The explicit construction of $\mathfrak{u}$ from the Chevalley basis builds toward 07.04.03 (Cartan involution), where the involution $\theta (X+iY)=X-iY$ is shown to be unique up to conjugacy, and appears again in 07.06.03 (root systems), since the compact real form is built from the root space decomposition. The foundational reason is that the Chevalley basis has integer structure constants, ensuring the real span closes under the bracket, and this is exactly the bridge from the algebraic data (root system) to the topological data (compact group). Putting these together, the compact real form identifies the complex algebra $\mathfrak{g}$ with the Lie algebra of a compact Lie group, generalising the passage from $\mathfrak{sl}(n,\mathbb{C})$ to $\mathfrak{su}(n)$.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Conjugacy of compact real forms). Any two compact real forms of a complex semisimple Lie algebra $\mathfrak{g}$ are conjugate by an inner automorphism: if ${\mathfrak{u}}_1,{\mathfrak{u}}_2$ are compact real forms, there exists $\phi \in \mathrm{Int}(\mathfrak{g})$ with $\phi ({\mathfrak{u}}_1)={\mathfrak{u}}_2$.

Theorem 2 (Weyl's unitary trick). Every finite-dimensional representation of a complex semisimple Lie algebra $\mathfrak{g}$ is completely reducible. The proof: restrict the representation to the compact real form $\mathfrak{u}$, lift to the compact Lie group $U$, average an inner product over $U$ via Haar measure to make the representation unitary, and conclude that every invariant subspace has an invariant complement.

Theorem 3 (Helgason's theorem on the Cartan decomposition). The Cartan decomposition $\mathfrak{g}=\mathfrak{u}\oplus i\mathfrak{u}$ satisfies: $\mathfrak{u}$ and $i\mathfrak{u}$ are orthogonal under the Killing form, $\kappa$ is negative definite on $\mathfrak{u}$, and $\kappa$ is positive definite on $i\mathfrak{u}$. The bilinear form $B_{\theta }(X,Y):=-\kappa (X,\theta (Y))$ is positive definite on $\mathfrak{g}$, where $\theta$ is the Cartan involution.

Theorem 4 (Compact forms from the Chevalley basis). For any Chevalley basis of $\mathfrak{g}$, the real span of $\{i{h}_j\}\cup \{e_{\alpha }-e_{-\alpha }\}\cup \{i(e_{\alpha }+e_{-\alpha })\}$ is a compact real form. All compact real forms arise this way (up to inner automorphism).

Theorem 5 (Classification via compact forms). The classification of real forms of a complex semisimple Lie algebra $\mathfrak{g}$ is equivalent to the classification of involutive automorphisms of $\mathfrak{g}$ modulo inner automorphisms. Each involution $\theta$ determines a real form ${\mathfrak{g}}_0=\{X\in \mathfrak{g}:\theta (X)=X\}$, and the compact real form corresponds to the involution with the most symmetric eigenvalue structure.

Theorem 6 (Maximal compact subgroup). Every connected semisimple Lie group $G$ has a maximal compact subgroup $K$, unique up to conjugation. If $G$ is the complex group associated to $\mathfrak{g}$, then $K$ has Lie algebra $\mathfrak{u}$ (the compact real form). The coset space $G/K$ is diffeomorphic to Euclidean space (the Cartan decomposition at the group level).

Synthesis. The compact real form is the foundational reason that the algebraic theory of complex semisimple Lie algebras connects to the topological theory of compact Lie groups: the Killing form becomes a Riemannian metric, representations become unitary, and cohomology becomes computable. The central insight is that the Chevalley basis --- with its integer structure constants --- ensures the real span closes to give a compact form, and this is exactly the bridge from root systems 07.06.03 to the representation theory of compact groups 07.07.01. Putting these together with Weyl's unitary trick, every finite-dimensional representation of $\mathfrak{g}$ is completely reducible because its restriction to the compact form is unitarisable, and the pattern generalises: the Cartan-Weyl classification 07.04.01 of complex semisimple Lie algebras induces the classification of compact Lie groups via compact real forms. The bridge is the exponential map carrying the negative-definite algebra to the compact group.

Full proof set [Master]

Proposition 1 (Weyl's unitary trick). Every finite-dimensional representation of a complex semisimple Lie algebra $\mathfrak{g}$ is completely reducible.

Proof. Let $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$ be a finite-dimensional representation. Choose a compact real form $\mathfrak{u}\subset \mathfrak{g}$ and let $U$ be the corresponding compact, simply connected Lie group with $\mathrm{Lie}(U)=\mathfrak{u}$.

The representation $\rho$ restricts to $\rho {|}_{\mathfrak{u}}:\mathfrak{u}\to \mathfrak{gl}(V)$, which integrates to a representation $\hat{\rho }:U\to \mathrm{GL}(V)$ of the compact group.

Choose any Hermitian inner product $⟨\cdot ,\cdot {⟩}_0$ on $V$ and average over $U$:

$$⟨v,w⟩:={\int }_U⟨\hat{\rho }(u)v,\hat{\rho }(u)w{⟩}_{0}d\mu (u)$$

where $\mu$ is the Haar measure on $U$ with $\mu (U)=1$. This inner product is $U$-invariant: $⟨\hat{\rho }(u)v,\hat{\rho }(u)w⟩=⟨v,w⟩$ for all $u\in U$.

If $W\subset V$ is a $\mathfrak{g}$-invariant subspace, it is in particular $\mathfrak{u}$-invariant, hence $U$-invariant. The orthogonal complement $W^{\perp }$ in the averaged inner product is also $U$-invariant, hence $\mathfrak{u}$-invariant, hence $\mathfrak{g}$-invariant (since $\mathfrak{g}=\mathfrak{u}\oplus i\mathfrak{u}$). So $V=W\oplus W^{\perp }$ as $\mathfrak{g}$-representations. By induction on dimension, $V$ decomposes as a direct sum of irreducibles. $\square$

Proposition 2 (The Cartan decomposition is orthogonal). The Killing form satisfies $\kappa (\mathfrak{u},i\mathfrak{u})=0$, is negative definite on $\mathfrak{u}$, and positive definite on $i\mathfrak{u}$.

Proof. Let $\theta$ be the Cartan involution with $\theta {|}_{\mathfrak{u}}=\mathrm{id}$ and $\theta {|}_{i\mathfrak{u}}=-\mathrm{id}$. Since $\theta$ is an automorphism, $\kappa (\theta X,\theta Y)=\kappa (X,Y)$ for all $X,Y\in \mathfrak{g}$.

For $X\in \mathfrak{u}$ and $Y\in i\mathfrak{u}$: $\kappa (X,Y)=\kappa (\theta X,\theta Y)=\kappa (X,-Y)=-\kappa (X,Y)$, so $\kappa (X,Y)=0$.

For $X\in \mathfrak{u}$: $\kappa (X,X)=\kappa (\theta X,\theta X)=\kappa (X,X)$ (no constraint from $\theta$). Negative definiteness follows from the construction in the key theorem (Step 4).

For $iX\in i\mathfrak{u}$ with $X\in \mathfrak{u}$: $\kappa (iX,iX)=-\kappa (X,X)>0$ (using $\kappa$ is the complexification of the real form). So $\kappa$ is positive definite on $i\mathfrak{u}$. $\square$

Connections [Master]

• Cartan-Weyl classification 07.04.01. The compact real form is a canonical real structure attached to each complex semisimple Lie algebra in the Cartan-Weyl classification. The classification of compact Lie groups is equivalent to the classification of complex semisimple Lie algebras via the correspondence $\mathfrak{g}\mapsto \mathfrak{u}$.

• Root systems 07.06.03. The compact real form is constructed from the root space decomposition of $\mathfrak{g}$ via the Chevalley basis. The root system $\Phi$ determines the compact real form up to inner automorphism, and the Weyl group acts on the compact form as the group generated by reflections in the unit sphere of $\mathfrak{u}$.

• Casimir element 07.06.10. On the compact real form, the Casimir operator acts as a negative scalar on each irreducible representation (since the Killing form is negative definite). The Casimir eigenvalue distinguishes irreducibles by their highest weight, and its negative definiteness on $\mathfrak{u}$ is the engine behind Weyl's complete reducibility theorem.

• Compact Lie group representation 07.07.01. The representation theory of the compact group $U$ (with $\mathrm{Lie}(U)=\mathfrak{u}$) is equivalent to the representation theory of the complex algebra $\mathfrak{g}$. The Peter-Weyl theorem 07.07.02 decomposes $L^2(U)$ into irreducible characters, indexed by the dominant weights of $\mathfrak{g}$.

Historical & philosophical context [Master]

Elie Cartan proved the existence of compact real forms in his 1914 paper Les groupes reels simples finis et continus (Ann. Sci. Ecole Norm. Sup. 31) ^[Cartan1914], classifying all real forms of complex simple Lie algebras. Hermann Weyl used the compact real form in his 1925--26 four-part paper Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen (Math. Z. 23--24) ^[Weyl1925] to prove complete reducibility of representations via the unitary trick, settling a question that had resisted purely algebraic methods. Helgason's 1978 monograph Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press) ^{[Helgason1978]} gives the definitive modern treatment, establishing the correspondence between compact real forms and Riemannian symmetric spaces of compact type.

The Cartan decomposition $\mathfrak{g}=\mathfrak{u}\oplus i\mathfrak{u}$ is the infinitesimal version of the polar decomposition $G=K\cdot \exp (\mathfrak{p})$ at the group level, where $K$ is the maximal compact subgroup with $\mathrm{Lie}(K)=\mathfrak{u}$. This decomposition is the foundation of the theory of symmetric spaces, developed by Cartan in the 1920s and systematised by Helgason.

Bibliography [Master]

@article{Cartan1914,
  author = {Cartan, Elie},
  title = {Les groupes reels simples finis et continus},
  journal = {Ann. Sci. Ecole Norm. Sup.},
  volume = {31},
  year = {1914},
  pages = {263--355},
}

@article{Weyl1925,
  author = {Weyl, Hermann},
  title = {Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. {I}, {II}, {III}, {IV}},
  journal = {Math. Z.},
  volume = {23--24},
  year = {1925--1926},
}

@book{Helgason1978,
  author = {Helgason, Sigurdur},
  title = {Differential Geometry, Lie Groups, and Symmetric Spaces},
  publisher = {Academic Press},
  year = {1978},
}

@book{Knapp2002,
  author = {Knapp, Anthony W.},
  title = {Lie Groups Beyond an Introduction},
  publisher = {Birkhauser},
  year = {2002},
  edition = {2nd},
}

@book{Humphreys1972,
  author = {Humphreys, James E.},
  title = {Introduction to Lie Algebras and Representation Theory},
  publisher = {Springer},
  year = {1972},
}