For six dimensional Lie algebra with non-zero Lie brackets defined as follow: $[e_{1}, e_{3}] = -e_{1}, [e_{1}, e_{6}] = -e_{2}, [e_{2}, e_{3}] = -e_{2}, [e_{2}, e_{4}] = e_{1}, [e_{2}, e_{5}] = e_{2}, [e_{4}, e_{5}] = -e_{4}, [e_{4}, e_{6}] = -2*e_{5}-e_{3}, [e_{5}, e_{6}] = -e_{6}$. What would be quotient algebra $\frac{\text{Nor}\left(w_{1}\right)}{w_{1}}$ for $w_{1}=a*e_{3}+e_{5}$ ? Where 'a' is arbitrary constant $\neq 0, 1$ . The expected answer is $\{e_{3}\}$
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What are all the $a$'s? – Tobias Kildetoft Apr 19 '16 at 07:57
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1What does that mean? Please try to write this as you would in math terms, rather than Maple notation. – Tobias Kildetoft Apr 19 '16 at 08:09
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The question is fully edited in simpler version. – IgotiT Apr 19 '16 at 08:24
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What is the normalizer of an element? You mean the normalizer of the subalgebra generated by that element? And similarly, you you mean to take the quotient by the subalgebra generated by the element, rather than just the element? – Tobias Kildetoft Apr 19 '16 at 08:29
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If $[w_{1}, w_{2}]= \lambda*w_{1}$ then we say $w_{2}\in \text{Nor}\left(w_{1}\right)$, where $\lambda$ being arbitrary constant. – IgotiT Apr 19 '16 at 08:32
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I mean quotient algebra of Normalizer. – IgotiT Apr 19 '16 at 08:37
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Right, so that is the normalizer of the subalgebra generated by the element. – Tobias Kildetoft Apr 19 '16 at 08:37
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The normalizer can be straightforwardly computed using Maple package Lie Algebra but problem is with its quotient algebra. – IgotiT Apr 19 '16 at 08:39
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I am not sure how the expected answer could be some element from the Lie algebra itself, as the quotient will not be a subset of this. Do you mean you expect that element to map to a generator of the quotient under the natural projection? – Tobias Kildetoft Apr 19 '16 at 08:42
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Please see research article [http://scitation.aip.org/content/aip/journal/jmp/33/10/10.1063/1.529907] at page 3592, in first column and 4th 2-dimensional sub-algebra for such construction. – IgotiT Apr 19 '16 at 11:18
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It says file not found. – Tobias Kildetoft Apr 19 '16 at 11:19
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"Group‐invariant solutions and optimal systems for multidimensional hydrodynamics", Journal of Mathematical Physics. – IgotiT Apr 19 '16 at 11:21
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Sorry, I don't really feel like looking through a paper on mathematical physics to try to understand what you are asking. – Tobias Kildetoft Apr 19 '16 at 11:23
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This paper concerns with classification of six dimensional Lie algebra. Anyway, thanks for your concern. – IgotiT Apr 19 '16 at 11:28