Real forms and covering groups

A real form of a complex Lie algebra g is a real Lie algebra g₀ whose complexification g₀ ⊗ℝ ℂ is isomorphic to g. Its real dimension equals the complex dimension of g. The real form specifies the real algebra; the integrating group supplies additional global topology. This reference links 19 real forms to their actual real-algebra records and selected connected groups.

The definition and classical classification are developed in Etingof, Sections 41.2–41.4. The complex-type index gives the family ranges and low-rank identifications.

Compact real forms in the atlas

A complex semisimple algebra has a compact real form, unique up to isomorphism. Its real rank is zero. The following table lists the 14 compact real-form records and their curated global integrations. Groups on the same row share a real algebra; they can have different centers and fundamental groups. Compact Sp(n) uses quaternionic dimension.

Compact real form	Complexification type	Curated groups
su(2)	A₁	SU(2), SO(3)
su(3)	A₂	SU(3), PSU(3)
su(4) ≅ so(6)	A₃	SU(4), SO(6), PSU(4)
su(5)	A₄	SU(5)
Compact sp(2) ≅ so(5)	B₂ = C₂	Sp(2), SO(5)
Compact sp(3)	C₃	Sp(3)
so(8)	D₄	Spin(8), SO(8), PSO(8)
so(10)	D₅	Spin(10), SO(10)
Compact e₆	E₆	Simply connected and adjoint E₆
Compact e₇	E₇	Simply connected and adjoint E₇
Compact e₈	E₈	E₈
Compact f₄	F₄	F₄
Compact g₂	G₂	G₂
su(2) ⊕ su(2)	A₁ ⊕ A₁	Spin(4), SO(4)

For existence, compact structure, and the central quotients, see Conrad–Landesman, Lectures 26–28 and Appendix V and Etingof, Sections 41 and 43. The last row is semisimple rather than simple.

Selected noncompact real forms

The five noncompact records emphasize matrix groups, spacetime signatures, and their covers. Real rank is the dimension of a maximal abelian subspace of the noncompact part of a Cartan decomposition. It differs from the rank of the complexified algebra: su(2,2) has real rank two and complexification of rank three. Knapp, Chapter VI, Section 4, develops restricted roots and these rank conventions.

Real form	Complexification	Selected integrations
sl(2,ℝ)	A₁	SL(2,ℝ), PSL(2,ℝ), universal cover
sl(3,ℝ)	A₂	SL(3,ℝ)
su(2,2) ≅ so(2,4)	A₃	SU(2,2), SO₀(2,4)
sp(4,ℝ) ≅ so(2,3)	B₂ = C₂	Sp(4,ℝ), SO₀(2,3)
sl(2,ℂ) viewed over ℝ	A₁ ⊕ A₁	SL(2,ℂ), SO₀(1,3)

Here the subscript 0 selects the identity component. Real Sp(4,ℝ) acts on a four-dimensional real symplectic space; it is different from compact Sp(2), despite their isomorphic complexified algebras.

Realification is a different operation

Start with the complex algebra sl(2,ℂ) and retain only real scalar multiplication. The resulting real algebra has dimension six and is simple over ℝ. Complexifying it produces sl(2,ℂ) ⊕ sl(2,ℂ), not a single A₁. This follows from gℝ ⊗ℝ ℂ ≅ g ⊕ conjugate(g); see Knapp, Theorem 6.105, for complex simple algebras within the real classification.

The Lorentz algebra and the compact algebra of Spin(4) therefore share a complexification but are distinct real forms. Their finite-dimensional complex modules use two highest-weight labels; the real group, signature, and descent condition must remain attached to those labels.

Check the global cover

SL(2,ℝ) retracts onto SO(2), so π₁ is ℤ. Its universal cover has infinite cyclic center. With central generator z lifting a π rotation, the projection to SL(2,ℝ) kills ⟨z²⟩; projection to PSL(2,ℝ) kills ⟨z⟩. Etingof, Section 43.2, supplies the polar decomposition; Brodzki and collaborators, Section 1, describes the universal cover.

The spin maps SU(2,2) → SO₀(2,4) and Sp(4,ℝ) → SO₀(2,3) are double covers with kernel {±I}. Both source groups have π₁ = ℤ; both targets have π₁ = ℤ × C₂. They are not universal covers. Their maximal compact subgroups determine these homotopy groups by Cartan decomposition. See Garrett, Sections 2.8 and 2.11, and Knapp, Chapter VI, Sections 3–4.

For spacetime use, start with the Lorentz spin map or the specified indefinite orthogonal map, then fix any further projective quotient or covering group. The 19 entries give selected real forms and integrations, not an exhaustive index of noncompact groups or unitary duals. The u(1) and u(2) reductive algebra records are also available, separately from this semisimple real-form index.