Timeline for Witt algebra, $\mathfrak{sl}(2,\mathbb{R})$, $\mathfrak{sl}(2,\mathbb{C})$ and Bosonic String Theory
Current License: CC BY-SA 4.0
7 events
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| Sep 9, 2021 at 21:12 | comment | added | ACuriousMind♦ | @AlessioFontanarossa I'm not sure why you think you're missing anything - why some authors might choose to view the algebras as real or complex depends on the context. E.g. you have to complexify if you're thinking about the quantum theory, since quantum theory is about complex representations, but classically there's no need to. It would probably help if you cited one of the claims from these authors explicitly and explain what your issue with it is. | |
| Sep 9, 2021 at 20:53 | comment | added | Alessio Fontanarossa | N2, part two When we have the two copy of $\mathfrak{sl}(2,\mathbb{R})$ for $L_{n}$ and $\bar{L}_{n}$, we are directly summing them or complexifying them? and why? I would answer "summing" (since $[L,\bar L]=0$), but in that case I would not obtain $\mathfrak{sl}(2,\mathbb{C})$ (as in the case $\mathfrak{su}(2)\oplus\mathfrak{su}(2)\sim\mathfrak{so}(4)$) as you have said and as is right . So, what am I missing? Thank you again a lot for your time. | |
| Sep 9, 2021 at 20:52 | comment | added | Alessio Fontanarossa | N2, part one <br/> Complexification. Indeed it is known that $\mathfrak{su}(2)\oplus\mathfrak{su}(2)\sim\mathfrak{so}(4)$ and also that $\mathfrak{su}(2)_{\mathbb{C}}\sim\mathfrak{sl}(2,\mathbb{R})_{\mathbb{C}}\sim\mathfrak{sl}(2,\mathbb{C})$. I understand that we can simbolically write the complexification of a real vector space $V_{\mathbb{C}}$ as a direct sum $V\oplus V$ (also without the $i$), where it is intended that we are using a definition for the multiplication for $i$. But in our case? | |
| Sep 9, 2021 at 20:49 | comment | added | Alessio Fontanarossa | I completely agree with your answers 1,3,4; that has been very helpful for me, so thank you a lot. I edit this comment before you edit the answer, so you don't have to spend too much time replying again. I have still 2 problems: N1 Why authors choose to use a real algebra for $L_{-1,0,1}$, saying that it is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$? Are there links with the fact that $SL(2,\mathbb{C})$ is simply connected and $SL(2,\mathbb{R})$ is not? | |
| Sep 9, 2021 at 16:27 | history | edited | ACuriousMind♦ | CC BY-SA 4.0 |
added 92 characters in body
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| Sep 9, 2021 at 9:29 | history | edited | ACuriousMind♦ | CC BY-SA 4.0 |
if I say we have to be careful about real/complex then we should actually be careful...
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| Sep 8, 2021 at 17:30 | history | answered | ACuriousMind♦ | CC BY-SA 4.0 |