Timeline for Universal property of induced representation
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 16, 2020 at 15:38 | vote | accept | Hans | ||
| Jun 16, 2020 at 15:38 | vote | accept | Hans | ||
| Jun 16, 2020 at 15:38 | |||||
| Jun 16, 2020 at 15:38 | vote | accept | Hans | ||
| Jun 16, 2020 at 15:38 | |||||
| Jun 16, 2020 at 12:29 | vote | accept | Hans | ||
| Jun 16, 2020 at 12:29 | |||||
| Jun 16, 2020 at 10:01 | answer | added | Vít Tuček | timeline score: 2 | |
| Jun 16, 2020 at 8:43 | answer | added | Bugs Bunny | timeline score: 2 | |
| Jun 15, 2020 at 18:24 | comment | added | Hans | Maybe I am overlooking something but it seems to me that one needs that $iE$ is reflexive or something in that direction. Is that true? | |
| Jun 15, 2020 at 17:42 | comment | added | dorebell | Oh sorry, I was being careless! The definition you give is naturally the right adjoint to restriction (in the category of continuous representations). However, since the category of continuous representations of a compact Lie group is semisimple (i.e. there’s always a unitary structure), you can apply Frobenius reciprocity to the duals and then dualize back to get the other universal property. This isn’t as natural as the other statement, because it’s relying crucially on semisimplicity - for more general groups, the right and left adjoints are indeed sometimes different. | |
| Jun 15, 2020 at 17:19 | comment | added | Hans | In the case of finite groups, induction is both left and right adjoint to restriction. Is that true in general? In the book I am looking at, they only prove one of these two. In particular, I only get a natural map $iE\to E$ from that theorem. What is the natural map $E\to iE$? | |
| Jun 15, 2020 at 17:11 | comment | added | dorebell | By unwinding the Frobenius reciprocity theorem, which says the functors of restriction and induction are adjoint, you’ll get exactly this sort of universal property. However, some care is needed - since this definition of induced representation imposes a continuity condition, it will only have the universal property among continuous representations. | |
| Jun 15, 2020 at 16:52 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals
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| Jun 15, 2020 at 16:49 | history | asked | Hans | CC BY-SA 4.0 |