Timeline for Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds
Current License: CC BY-SA 4.0
26 events
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| Aug 19, 2021 at 18:52 | history | edited | wonderich | CC BY-SA 4.0 |
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| Aug 19, 2021 at 16:45 | comment | added | user43326 | The definition of a representation of G into H, or Rep[G,H] is a equivalence class of a homomorphisms from G to H modulo conjugation by an element of H. I wasn't talking about induced map $R_O(H)\to R_O(G)$, sorry for confusion. | |
| Aug 18, 2021 at 19:02 | comment | added | LSpice | @user43326, is it obvious that two homomorphisms that do the same thing on representations differ by conjugation? I'm not even sure I see it for $G$ compact. | |
| Aug 18, 2021 at 16:56 | comment | added | user43326 | Two homomorphism which are same representations are related to each other by a conjugation by an element of a target group $G_2$. Let's call this element $g$. There is a homotopy between $g$ and $e$ (the unit element of $G_2$, which should give a "continuous deformation of embedding", but I am not sure exactly what you mean by "continuous deformation of embedding". | |
| Aug 18, 2021 at 13:20 | comment | added | wonderich | I update my question and clarify that this constraint in the first paragraph. (you can illuminate or vote up to help on getting an answer.) | |
| Aug 18, 2021 at 13:19 | history | edited | wonderich | CC BY-SA 4.0 |
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| Aug 18, 2021 at 13:18 | comment | added | wonderich | yes I am asking the particular embedding(s) that agree with my mapping between representations. | |
| Aug 18, 2021 at 11:24 | comment | added | Fernando Muro | Are you maybe asking about the possibility of constructing a Lie group embedding satisfying certain conditions? | |
| Aug 17, 2021 at 21:21 | comment | added | wonderich | This is motivated by some gauge theory problems in physics. The conditions given here are also motivated by that representation theory in gauge theory. My post shows what is typically given. I must use the representation theory to specify the embedding (perhaps with more data, which is part of the question: how many distinct classes there are with the given representation map?). | |
| Aug 17, 2021 at 21:10 | comment | added | Fernando Muro | There's no ambiguity at all even if you don't talk about representation theory. I guess your question is not well explained. | |
| Aug 17, 2021 at 21:06 | comment | added | wonderich | If you think these give the same embedding, or you could rule out them, please post an answer. Thank you! | |
| Aug 17, 2021 at 21:03 | comment | added | wonderich | There you see that there could be at least two ways to look at the embedding of $SU(5) \subset Spin(10)$. There could be more ways for example. $$ \mathbf 5 \oplus \overline{\mathbf{10}} \oplus \mathbf 1 \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10).$$ $$ \mathbf 1 \oplus \mathbf{10} \oplus \overline{\mathbf{5}} \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10).$$ $$ \mathbf{5} \oplus \overline{\mathbf{5}} \text{ in } SU(5) \mapsto \mathbf{10} \text{ in } Spin(10).$$ or others. | |
| Aug 17, 2021 at 21:01 | comment | added | wonderich | @Fernando Muro, thanks for your input - but please look at this example -- mathoverflow.net/q/401946/27004 if we do not specify the representation, there could be ambiguity. My question is exactly about to give that representation maps, how to "specify the image of each point in the small group into the larger group"? | |
| Aug 17, 2021 at 20:36 | review | Close votes | |||
| Aug 22, 2021 at 3:04 | |||||
| Aug 17, 2021 at 20:20 | comment | added | Fernando Muro | I mean, the answer to your question is trivial in the sense that you only need to specify the image of each point in the small group into the larger group. Then it has to satisfy the obvious properties. What you explain later is totally unclear to me. You don't need representations to specify an embedding. | |
| Aug 17, 2021 at 17:44 | history | edited | wonderich | CC BY-SA 4.0 |
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| Aug 17, 2021 at 17:42 | comment | added | wonderich | I should clarify that a continuous deformation of embedding may be regarded as the same class of embedding. A discrete deformation of embedding may be regarded as the different class of embedding. | |
| Aug 17, 2021 at 17:41 | comment | added | wonderich | "the statement that a homomorphism of Lie groups is completely determined by its derivative at the identity" --> actually I am more looking into how to classify possible distinct embeddings if the data of eq.(1) is given. Maybe there are several ways to embed even we fix the data of eq.(1). But I do not know enough math to help me find the precise way to solve or classify this embedding. | |
| Aug 17, 2021 at 17:34 | comment | added | wonderich | thanks so much, I voted up your comment, see also my concrete example: mathoverflow.net/q/401946/27004 | |
| Aug 17, 2021 at 17:27 | comment | added | LSpice | (But notice that even completely understanding the resulting map $\operatorname{Rep}(G_2) \to \operatorname{Rep}(G_1)$ cannot determine the map $G_1 \to G_2$, since it is insensitive to conjugation on the target by $\operatorname{Aut}(G_2)$ (and hence to conjugation on the source by $G_1$).) | |
| Aug 17, 2021 at 17:26 | comment | added | LSpice | Maybe you're looking for the statement that a homomorphism of Lie groups is completely determined by its derivative at the identity, as a map of Lie algebras? | |
| Aug 17, 2021 at 16:59 | comment | added | wonderich | What I want to know is like a precise map, for a simple example like the embedding between two vector spaces or two flat manifolds $$\mathbb{R}^2 \mapsto \mathbb{R}^3,$$ then I can uniquely specify the embedding if I know the maps between two-independent basis vectors ${e_1,e_2}$ of $\mathbb{R}^2$ to two-independent basis vectors of $\mathbb{R}^3.$ in the form of their basis ${e_1',e_2',e_3'}$. I wanted to know how to specify or enumerate the embedding given the way how their representations are mapped. Thank you! | |
| Aug 17, 2021 at 16:56 | history | edited | wonderich | CC BY-SA 4.0 |
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| Aug 17, 2021 at 16:53 | comment | added | wonderich | yes, in the sense of geometry embedding. please help me to clarify. (I will ask an example next.) thanks! | |
| Aug 17, 2021 at 16:51 | comment | added | Fernando Muro | You're asking about how to define a map | |
| Aug 17, 2021 at 16:47 | history | asked | wonderich | CC BY-SA 4.0 |