- #1

aalma

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- 1

- Homework Statement
- See below

- Relevant Equations
- We recall that the fundamental group of ##SL(2,R)## is ##Z## and define by ##G## the universal covering group for ##SL(2,R)##.

Show that any complex representation of G, ##ρ : G →GL(n,C)##, factors through ##SL(2,R)## (that is ##ρ## is the composition ##G →SL(2,R) →GL(n,C)##).

Hint: Any representation ##ρ## gives rise to a representation of the Lie algebra ##sl(2, R) → gl(n,C)##. This extends to ##sl(2,C) → gl(n,C)## that can be uniquely lifted to ##SL(2,C) → GL(n,C)##. The composition with ##G → SL(2,R) → SL(2,C)## should coincide with ##ρ##.

I am not quite sure of how this works, i.e. of what exactly I need to do with the hint.

Any explanantion would be helpful!

Any explanantion would be helpful!