Timeline for Automorphisms of the Lie algebras $\mathfrak{sl}(2,R)$ and $\mathfrak{su}(2)$

6 events

when toggle format	what		by	license	comment
Feb 1, 2019 at 11:42	comment	added	Ben McKay		@Abenthy: yes, because $SL(2,\mathbb{R})$ is connected. Its automorphisms clearly give automorphisms of its Lie algebra, and these arise from $\mathbb{P}GL(2,\mathbb{R})$. Each automorphism of $SL(2,\mathbb{R})$ commutes with the exponential map, so is determined near the identity element by the automorphism of the Lie algebra. By analyticity, and connectedness of $SL(2,\mathbb{R})$, the automorphism is determined uniquely, and so must be that element of $\mathbb{P}GL(2,\mathbb{R})$.
Feb 1, 2019 at 9:55	comment	added	Abenthy		@Ben McKay: Does this also imply that the automorphisms of the Lie group $\text{SL}(2,\mathbb{R})$ are precisely the ones given by conjugation with elements of $\text{GL}(2,\mathbb{R})$?
Aug 24, 2017 at 14:09	comment	added	LSpice		I think it might be fair to say that the lesson you are imparting is that it's easier to understand automorphisms of complex reductive Lie algebras that preserve real structures than to understand automorphisms of real Lie algebras directly.
Aug 24, 2017 at 13:56	vote	accept	Javier
Aug 24, 2017 at 13:38	history	edited	Ben McKay	CC BY-SA 3.0	added su(2)
Aug 24, 2017 at 12:56	history	answered	Ben McKay	CC BY-SA 3.0