$\widetilde{\mathrm{SL}(2,\mathbb R)}$Simply connected group integrating sl(2,R)

The simply connected covering Lie group of SL(2,R), or equivalently of PSL(2,R).

• K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: the universal covering group of SL(2,R)Open source

3

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• 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

1

• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

Z

• K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: infinite cyclic center and absence of nontrivial compact subgroupsOpen source

0

• Lie Groups and Lie AlgebrasSection 3.2: universal covering Lie groupsOpen source

No

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

Yes

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

No

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

Yes

• 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 groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

Simply connected real Lie group; not a matrix group

• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
• K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: structure of the universal coverOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.46: Iwasawa decompositionOpen source
• Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

sl(2, R)

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

Trivial

• K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: infinite cyclic center and absence of nontrivial compact subgroupsOpen source

The lifted Iwasawa factors identify its manifold with R³. Let z be the central element represented by a lift of a rotation through π in SL(2,R). Then the projection to SL(2,R) has kernel ⟨z²⟩, while the projection to PSL(2,R) has kernel ⟨z⟩.

• K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: structure of the universal coverOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.46: Iwasawa decompositionOpen source
• Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

Every finite-dimensional complex representation factors through SL(2,R). Consequently this simply connected Lie group has no faithful finite-dimensional matrix representation; its infinite cyclic center is not detected faithfully in that category.

• Lie Groups and Lie AlgebrasRemark 36.7, printed page 196; Exercise 11.20: finite-dimensional representations and nonlinearity of the universal coverOpen source

algebra:sl-2-r

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

real-form:sl-2-r

• Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

map:universal-sl-2-r-to-sl-2-r

• K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: structure of the universal coverOpen source
• Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.46: Iwasawa decompositionOpen source
• Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source