0

Currently I am studying lie theory and differentiable manifolds together, and I encountered a question regarding the image of exponential map from lie algebra of $SL(2, \mathbb{R})$ on to $SL(2, \mathbb{R})$. From what I know, lie algebra of $SL(2, \mathbb{R})$ is just 2 by 2 real matrix with trace 0. Now I am stuck at the question where I need to determine the image of exponential map for lie group $SL(2, \mathbb{R})$. (i.e. what is an image of lie algebra $sl(2, \mathbb{R})$ onto $SL(2, \mathbb{R})$?)

My idea was just to consider the basis of $SL(2, \mathbb{R})$ and chug some result out, but so far they are unsuccessful. Does anyone know how to approach this question? Thank you!

엄익훈
  • 115
  • 7
  • It's not really clear to me what you are asking. Are you asking for the image of a single element of the Lie algebra or are you asking what is the image of the exponential map as a subset of $SL(2,\mathbb{R})$? – Callum Feb 04 '24 at 13:24
  • Hello, the question is asking for the image of exponential map as a subset of SL(2, R), and if possible, how can I classify the image of exponential map – 엄익훈 Feb 04 '24 at 13:27
  • The only way I know off the top of my head is that an element is in the image of the exponential map if and only if it is the square of another element. See here for example (that example is for $GL(n,\mathbb{R})$ but the same argument should work). – Callum Feb 04 '24 at 13:40
  • nvm easily solved by case by case exhaustion on possible eigenvalues of SL(2, R):) Maybe I just needed more time to see how to tackle this problem – 엄익훈 Feb 04 '24 at 17:23
  • Compare this answer: https://math.stackexchange.com/a/3239741/96384 – Torsten Schoeneberg Feb 04 '24 at 22:38

0 Answers0