Timeline for A solvable Lie algebra
Current License: CC BY-SA 3.0
8 events
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| Apr 2, 2015 at 13:38 | history | edited | Eber | CC BY-SA 3.0 |
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| Apr 2, 2015 at 13:00 | history | edited | Eber | CC BY-SA 3.0 |
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| Apr 2, 2015 at 12:50 | history | edited | Eber | CC BY-SA 3.0 |
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| Apr 2, 2015 at 12:00 | comment | added | Dietrich Burde | The Lie group $Sol_{2n-1}$ and its Lie algebra are "very known", also in geometry. For example, $Sol_{2n-1}$ has a quadratic Dehn function for $n\ge 3$ (C. Pittet). | |
| Apr 2, 2015 at 10:05 | comment | added | YCor | I know this Lie algebra as the direct product of a 1-dimensional abelian Lie algebra and a $2n-1$-dimensional Lie algebra called $\mathfrak{sol}_{2n-1}$, usually defined as the semidirect product of an $n$-dimensional ideal by the $(n-1)$-dimensional of diagonal matrices of trace 0. (PS: in my previous comment I mean "$n$ weights", not "$n$-weights".) | |
| Apr 2, 2015 at 5:27 | comment | added | YCor | If $n=2$ it's not well-defined (because $[x_1,x_2]=y_1$ and $[x_2,x_1]=y_2$ is absurd), so assume $n\ge 3$. Then set $X_i=x_i+y_{i-1}$. Then in the new basis $X_1,\dots,X_n,y_1,\dots,y_n$, the nonzero brackets are $[X_i,y_i]=-y_i$ and $[X_i,y_{i-1}]=y_{i-1}$. This is a semidirect product of two $n$-dimensional abelian Lie algebras, the first acting on the second with $n$-weights $w_1\dots,w_n$, which are dependent with only one relation ($\sum w_i=0$). Hence unlike what you claim, its center is 1-dimensional, generated by $\sum_i X_i=(\sum_i x_i+\sum_i y_i)$. | |
| Apr 2, 2015 at 2:35 | review | First posts | |||
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| Apr 2, 2015 at 2:32 | history | asked | Eber | CC BY-SA 3.0 |