Timeline for Mackey theory for semidirect products: equivalence between constructions for modules
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 3, 2020 at 19:46 | vote | accept | Angelo Lucia | ||
| Jun 3, 2020 at 0:05 | answer | added | LSpice | timeline score: 3 | |
| Jun 2, 2020 at 23:22 | comment | added | Angelo Lucia | @LSpice You are right, I made a mistake: the skew-product I wrote did not make any sense. I have added some more specific references, and clarified the meaning of the arrows as you suggested. Since my construction was flawed, I have modified my answer so that you can post your both comments as an answer. | |
| Jun 2, 2020 at 23:18 | history | edited | Angelo Lucia | CC BY-SA 4.0 |
Corrected my mistake, and reformulated the question in the light of it. Also added the required references and make some suggested clarifications.
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| Jun 2, 2020 at 5:11 | comment | added | LSpice |
($(n \rtimes h) \otimes_{\mathbb C(G_p)} (z \otimes_{\mathbb C} v) \mapsto p(n)z \otimes_{\mathbb C} (h \otimes_{\mathbb C(H_p)} v)$ and $(1 \rtimes h) \otimes_{\mathbb C(G_p)} (z \otimes_{\mathbb C} v) \leftarrow z \otimes_{\mathbb C} (h \otimes_{\mathbb C(H_p)} v)$ (stuck with $\leftarrow$ because we don't seem to have \mapsfrom).)
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| Jun 2, 2020 at 5:06 | comment | added | LSpice | If I'm right to look at Folland, Theorem 6.38, and Etingof et al., Section 4.26, then the two constructions they describe, in 'module language', are $\mathbb C(G) \otimes_{\mathbb C(G_p)} (\mathbb C_p \otimes_{\mathbb C} \mathcal H_\sigma)$ (where the $N$ piece of $G_p = N \rtimes H_p$ acts only on $\mathbb C_p$, and the $H_p$ piece acts only on $\mathcal H_\sigma$); and $\mathbb C_p \otimes_{\mathbb C} (\mathbb C(H) \otimes_{\mathbb C(H_p)} \mathcal H_\sigma)$ (similar action convention). It is clear that these are the same construction. | |
| Jun 2, 2020 at 5:01 | comment | added | LSpice | Also, (1) where do these two constructions occur? (The closest I can find is Folland, Theorem 6.38; Etingof et al., Section 4.26.) (2) Where are the parentheses in $\mathbb C_p \rtimes \mathbb C(H) \otimes_{\mathbb C(H_p)} \mathcal H_\sigma$? (3) In the notation $a\phi_h(b)$, I think that $b$ is a complex number, but $\phi_h$ is an automorphism of $N$; so what does $\phi_h(b)$ mean? | |
| Jun 2, 2020 at 4:44 | comment | added | LSpice | Although the conceptual meaning is pretty clear from the context, it may be worth clarifying that your arrows between Hilbert spaces are "construct from" arrows, not Hilbert-space morphisms. | |
| Jun 2, 2020 at 4:43 | history | edited | LSpice | CC BY-SA 4.0 |
Name of this paper
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| Jun 2, 2020 at 2:46 | history | asked | Angelo Lucia | CC BY-SA 4.0 |