Let O(n), SO(n), U(n), Sp(n, F), Sp(n) be the (real) orthogonal group, the (real) special orthogonal group, the West group, the Sin group and the tight Sin group, respectively. · Proof U(n) = O(2n) Sp(n, R), Sp(n) = U(2n) Sp(n, C). - Are O(n), SO(n), U(n), Sp(n, C), Sp(n) tightly connected? Prove your conclusion. - What are the universal covering groups of SO(n) and U(n)?
Let O(n), SO(n), U(n), Sp(n, F), Sp(n) be the (real) orthogonal group, the (real) special orthogonal group, the West group, the Sin group and the tight Sin group, respectively. · Proof U(n) = O(2n) Sp(n, R), Sp(n) = U(2n) Sp(n, C). - Are O(n), SO(n), U(n), Sp(n, C), Sp(n) tightly connected? Prove your conclusion. - What are the universal covering groups of SO(n) and U(n)?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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These questions are from a branch of mathematics known as Lie group theory.
![3.
Let O(n), SO(n), U(n), Sp(n, F), Sp(n) be the (real) orthogonal group, the (real) special orthogonal group,
the West group, the Sin group and the tight Sin group, respectively.
- Proof U(n) = O(2n) Sp(n, R), Sp(n) = U(2n) Sp(n, C).
- Are O(n), SO(n), U(n), Sp(n, C), Sp(n) tightly connected? Prove your conclusion.
- What are the universal covering groups of SO(n) and U(n)?
- Find the maximal subring groups, root systems and Dynkin diagrams of Sp(n).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F882ce437-6984-48a2-b1c1-0c70a26630a5%2F9bbd946c-e71c-4a86-a33d-e0ee580f7ac0%2Fnt3ifjb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3.
Let O(n), SO(n), U(n), Sp(n, F), Sp(n) be the (real) orthogonal group, the (real) special orthogonal group,
the West group, the Sin group and the tight Sin group, respectively.
- Proof U(n) = O(2n) Sp(n, R), Sp(n) = U(2n) Sp(n, C).
- Are O(n), SO(n), U(n), Sp(n, C), Sp(n) tightly connected? Prove your conclusion.
- What are the universal covering groups of SO(n) and U(n)?
- Find the maximal subring groups, root systems and Dynkin diagrams of Sp(n).
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