Simple Lie algebras and compact global forms

Finite-dimensional simple Lie algebras over ℂ fall into four classical families and five exceptional types. This classification fixes an infinitesimal complex algebra. A real Lie group additionally requires a real form and a global form. Keeping these choices separate makes dimension, topology, and representation statements unambiguous.

For example, the complex algebra sl(2,ℂ) has the compact real form su(2) and the split real form sl(2,ℝ). The compact algebra integrates to both SU(2) and SO(3), whose fundamental groups and allowed representations differ. Etingof, Section 41, describes real forms; Section 43 explains compact global forms.

The nine simple types

The dimension column is the complex dimension of the algebra, also the real dimension of its compact real form. Cₘ denotes a cyclic group of order m; 1 denotes the trivial group. The center belongs to the listed simply connected compact group.

Type	Rank convention	Dimension	Compact simply connected group	Center
Aᵣ	r ≥ 1	r(r + 2)	SU(r + 1)	Cᵣ₊₁
Bᵣ	r ≥ 2	r(2r + 1)	Spin(2r + 1)	C₂
Cᵣ	r ≥ 3 here	r(2r + 1)	Sp(r), compact	C₂
Dᵣ	r ≥ 4	r(2r − 1)	Spin(2r)	C₄ for odd r; C₂ × C₂ for even r
E₆	6	78	E₆, simply connected	C₃
E₇	7	133	E₇, simply connected	C₂
E₈	8	248	E₈	1
F₄	4	52	F₄	1
G₂	2	14	G₂	1

The nonredundant rank conventions appear in Knapp, Theorem 6.105. The center table is supported by Conrad, pages 2–4. The family pages also provide Coxeter and dual Coxeter numbers, using long-root squared length 2, with locators to Etingof, Section 32.4.

Low ranks and repeated names

The rank bounds prevent duplicate entries for isomorphic complex simple algebras. They do not forbid writing C₂ in a calculation. A change between B₂ and C₂ conventions must account for the conventional Dynkin-node labeling.

Algebra identity	Group interpretation and destination
A₁ = B₁ = C₁	SU(2) ≅ Spin(3) ≅ compact Sp(1). SO(3) is its central quotient.
B₂ = C₂	Sp(2) ≅ Spin(5); SO(5) is the quotient by {±I}.
D₃ = A₃	SU(4) ≅ Spin(6); its vector cover has target SO(6).
D₂ = A₁ ⊕ A₁	The complex algebra is semisimple, not simple. Spin(4) ≅ SU(2) × SU(2).
so(2,ℂ) is abelian	Its compact integration is the circle group; it is outside the simple classification.

These identifications are recorded in Conrad’s center handout, page 4. The global-group distinction matters: SO(4) is the quotient of Spin(4) by the diagonal C₂, rather than its entire center.

From a compact algebra to a global group

For a compact simple real algebra, let G̃ be its simply connected compact group. Every connected group with that algebra has the form G̃/Γ, where Γ is a subgroup of the finite center. Its fundamental group is Γ. An irreducible representation of G̃ descends precisely when Γ acts trivially. See Etingof, Section 43.1.

The chain SU(4) → SO(6) → PSU(4) illustrates two different kernels inside C₄. The first quotient removes {±I}; the full adjoint quotient removes all of C₄. The corresponding representation conditions differ despite the common algebra.

Distinct subgroups of a center need not produce distinct abstract groups. For Spin(8), triality permutes the three order-two central subgroups. Their quotients are abstractly isomorphic, while the specified covering maps distinguish vector and half-spin actions. PSO(8) instead removes the full C₂ × C₂ center. Conrad–Landesman, Remark T.2.1, p. 262, explains this automorphism action.

Choose the record for the question

Use a family for parameter ranges and root-system conventions, an algebra for local structure, a real form for signature and real rank, and a group for center, topology, and representation descent. Open a map when the kernel or embedding affects the calculation.

A complex group also has an underlying real Lie group. In particular, SL(2,ℂ) has real dimension six; its real algebra complexifies to A₁ ⊕ A₁. It is not another three-dimensional real form of A₁. Continue with real forms and covering groups for this distinction and the noncompact reference entries.