Timeline for How to define the determinant of a morphism between graded Lie algebras?
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| Mar 7, 2015 at 10:55 | comment | added | YCor | Anyway I'm puzzled by your question when you say "similar to the Euclidean case": even if $g_1,g_2$ are abelian of the same dimension, you don't have any canonical definition of determinant for a linear mapping $g_1\to g_2$. You need to fix some element in the maximal exterior power of each $g_i$ (which is 1-dimensional) in order to normalize. | |
| Mar 2, 2015 at 13:14 | comment | added | Changyu Guo | @YCor:In particular, such a Lie algebra admit a group operation * by the BCH formula so that g, with *, becomes a Carnot group of homogenuous dimension $Q=\sum_{i=1}^s i\dim V_i$. | |
| Mar 2, 2015 at 13:09 | comment | added | Changyu Guo | @YCor: I am sorry that I do not know there are many different notions of this concept. I followed the lecture notes by Enrico, which can be found here sites.google.com/site/enricoledonne/lecture_notes, the stratification of step $s$ means that $g=V_1\oplus V_2\cdots\oplus V_s$, with $[V_j,V_1]=V_{j+1}$ for $1\leq j\leq s-1$ and $V_s\neq \{0\}$. | |
| Mar 2, 2015 at 9:33 | comment | added | YCor | Could you provide your definition of stratified/graded Lie algebras? graded in positive integers? even in this context, there are many different non-equivalent definitions, especially some people make the requirement that the elements of degree 1 generate the Lie algebra. | |
| Mar 2, 2015 at 7:35 | history | asked | Changyu Guo | CC BY-SA 3.0 |