Timeline for Algorithm to determine whether there is an injective homomorphism between two Lie algebras
Current License: CC BY-SA 3.0
14 events
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| Nov 29, 2017 at 14:34 | comment | added | Nicola Ciccoli | Woops, sorry Josè, I missed your computations about the Killing form when writing my edit. | |
| Nov 29, 2017 at 13:24 | vote | accept | RubenM | ||
| Nov 29, 2017 at 13:24 | |||||
| Nov 29, 2017 at 9:01 | comment | added | Paul Levy | Yes, this modified answer shows that using an algorithm is the wrong way to understand the question. First understand the Lie algebras, then ask whether you really need to use a computer. | |
| Nov 28, 2017 at 23:18 | history | edited | José Figueroa-O'Farrill | CC BY-SA 3.0 |
added 279 characters in body
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| Nov 28, 2017 at 19:22 | history | edited | José Figueroa-O'Farrill | CC BY-SA 3.0 |
added a partial solution to the problem based on new information supplied by the OP.
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| Nov 28, 2017 at 17:26 | comment | added | RubenM | I've added the Lie algebras to my post. Sorry I didn't do it earlier. | |
| Nov 28, 2017 at 9:19 | comment | added | Nicola Ciccoli | I fail to see how the Levi decomposition of the $\mathfrak g$ should be related to the Levi decomposition of $\mathfrak h$. By identifying $\mathfrak h$ with a Lie subalgebra in $\mathfrak g$ it is clear that the Levi decomposition of the whole algebra does not bring much information on the LEvi decomposition of the subalgebra (as the example of $\mathfrak g=\mathfrak{gl}_n$ shows...) | |
| Nov 28, 2017 at 0:00 | comment | added | José Figueroa-O'Farrill | @YCor The OP claims that the structure constants of the Lie algebras in question are known. So my guess is that this is a very explicit calculation and one might be able to say more if one saw the Lie algebras explicitly. | |
| Nov 27, 2017 at 23:54 | comment | added | YCor | @LSpice I agree that for such questions, "complex" is a bad (but usual) shortcut for "algebraically closed of characteristic zero". The ground field question is important for input purposes as well. For such an algorithmic question, two fields come into play: the field of definition (which should be computable and in particular countable) and the field over which we can to determine whether there is a point (and for this it is reasonable to start with an algebraic closure, since asking about a subfield such as reals or p-adic, etc, is a harder elaboration) | |
| Nov 27, 2017 at 23:35 | comment | added | LSpice | Even to talk about the Levi decomposition in general, we'd want to be in characteristic 0, which gets back to @YCor's comment about the ground field. However, maybe it's only $p$-adic folks like myself who don't see "Lie algebra" and "complex Lie algebra" as synonymous. :-) | |
| Nov 27, 2017 at 22:50 | comment | added | YCor | Yes this is my point, it's a hard case (I don't claim the hardest). It's about an algorithm with input a pair of Lie algebras, so I'm not sure what you mean by "see the Lie algebra in question". | |
| Nov 27, 2017 at 22:39 | comment | added | José Figueroa-O'Farrill | I don't understand the "even if they're nilpotent". That seems to me to be the hardest case, since this is when, as you point out, the Levi decomposition does not give you a lot of information. One would have to see the Lie algebras in question to be able to answer the question. | |
| Nov 27, 2017 at 22:06 | comment | added | YCor | Even if $\mathfrak{h}$ and $\mathfrak{g}$ are nilpotent (in which case the Levi decomposition is trivial) I'm really pessimistic about a naive approach. | |
| Nov 27, 2017 at 22:02 | history | answered | José Figueroa-O'Farrill | CC BY-SA 3.0 |