Simple Lie type Dᵣ
Complex simple Lie type Dᵣ with dimension, normalized Coxeter data, compact cover, and its center.
Parameter domain
Classification of complex simple type; compact center and global-form rule; selected concrete real forms.
| Parameter | Allowed values |
|---|---|
| r | integer r ≥ 4 |
Reference properties
| Property | Value | Evidence |
|---|---|---|
| Rank | integer r ≥ 4 | Sources |
| Complex algebra dimension | r(2r − 1) | Sources |
| Coxeter number h | 2r − 2 | Sources |
| Dual Coxeter number h∨ | 2r − 2 | Sources |
| Compact simply connected group | Spin(2r) | Sources |
| Center of compact simply connected group | C₄ for r odd; C₂ × C₂ for r even | Sources |
| Root-length convention | Long roots have squared length 2 | Sources |
Algebra, real form, global group
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. [sources]
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. [sources]
Representation convention
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. [sources]
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. [sources]
Classical real forms
The real forms are so(p,q), p + q = 2r, and so*(2r). There are low-rank coincidences, including so*(8) ≅ so(6,2). [sources]
Rank conventions and coincidences
r = 1: so(2,C) is abelian, not simple. [sources]
r = 2: D₂ = A₁ ⊕ A₁ is semisimple, not simple. [sources]
r = 3: D₃ = A₃; its compact simply connected group is Spin(6) ≅ SU(4). [sources]
r = 4: Triality permutes the three nonidentity central elements of Spin(8), and hence the three order-two quotient kernels. [sources]
Related objects and maps
- Concrete algebra record
- Concrete algebra record
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
so(2r, C), with integer r ≥ 4.
- 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
family:lie-d#definitionRank
integer r ≥ 4
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
family:lie-d#rankComplex algebra dimension
r(2r − 1)
- 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
Premises: Rank r · Coxeter number h in Section 32.4
family:lie-d#dimensionCoxeter number h
2r − 2
- Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source
family:lie-d#coxeter-numberDual Coxeter number h∨
2r − 2
- Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source
family:lie-d#dual-coxeter-numberCompact simply connected group
Spin(2r)
- 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
family:lie-d#compact-coverCenter of compact simply connected group
C₄ for r odd; C₂ × C₂ for r even
- Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
family:lie-d#centerRoot-length convention
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
family:lie-d#root-normalizationAlgebra, real form, global group
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
family:lie-d#objects-1Algebra, real form, global group
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
family:lie-d#objects-2Representation convention
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
family:lie-d#representations-1Representation convention
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
family:lie-d#representations-2Classical real forms
The real forms are so(p,q), p + q = 2r, and so*(2r). There are low-rank coincidences, including so*(8) ≅ so(6,2).
- 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
family:lie-d#real-forms-1Rank conventions and coincidences
r = 1: so(2,C) is abelian, not simple.
- 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
family:lie-d#low-rank-1Rank conventions and coincidences
r = 2: D₂ = A₁ ⊕ A₁ is semisimple, not simple.
- 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
family:lie-d#low-rank-2Rank conventions and coincidences
r = 3: D₃ = A₃; its compact simply connected group is Spin(6) ≅ SU(4).
- 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
family:lie-d#low-rank-3Rank conventions and coincidences
r = 4: Triality permutes the three nonidentity central elements of Spin(8), and hence the three order-two quotient kernels.
- 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
family:lie-d#low-rank-4Concrete algebra record
algebra:d-4-c
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
family:lie-d#relation-1Concrete algebra record
algebra:d-5-c
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
family:lie-d#relation-2