$\mathrm{A}_{r}$Simple Lie type Aᵣ

sl(r + 1, C), with integer r ≥ 1.

• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

integer r ≥ 1

• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

r(r + 2)

• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
• Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source

r + 1

• Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source

r + 1

• Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source

SU(r + 1)

• Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
• Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source

Cᵣ₊₁

• Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source

Long roots have squared length 2

• Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source

A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.

• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
• Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.

• Conrad and Landesman · Compact Lie groupsAppendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ trialityOpen source

Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.

• Conrad and Landesman · Compact Lie groupsLectures 27–28: highest-weight classification and fundamental representationsOpen source
• Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotientsOpen source

The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.

• Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source

For n = r + 1 the real forms are su(p,q), p + q = n; sl(n,R); and su*(2m) = sl(m,H) when n = 2m, with low-rank coincidences understood.

• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.105, printed page 362: nonredundant real-form classification and coincidencesOpen source

r = 1: A₁ = B₁ = C₁. The canonical family entry is A.

• Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
• Conrad and Landesman · Compact Lie groupsAppendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ trialityOpen source

algebra:a-1-c

• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

algebra:a-2-c

• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

algebra:a-3-c

• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

algebra:a-4-c

• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source