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I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there is a natural way to speak of the determinant of $A$ similar to the Euclidean case, namely in the Euclidean case, a linear mapping $A:\mathbb{R}^n\to \mathbb{R}^n$ can be identified with a matrix and so the determinant can be defined as the determinant of the matrix (represented in the global coordinates).

Here the difficulty for me is the following: $g_1$ and $g_2$ are not necessarily the same Lie algebra and so I cannot jsut fix an orthogonal norm basis of each Lie aglebra and use the determinant of the corresponding matrix, since the representation matrix will then depend on the basis for $g_1$ and $g_2$ and is not invariant under change of basis.

Note also that $A$ is not necessarily an isomorphism of the Lie algebra, in which case, one could associate a volume measure for the Lie algebras and define the determinant of $A$ as volume derivatives. (In the Riemannian manifold case, one can use the invariant determinant for the differential $Df(p):T_pM\to T_qN$ of a smooth map $f:M\to N$, namely regard the differential as a tensor field).

References, suggestions and comments are greatly appreciated!

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  • $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Mar 2, 2015 at 9:33
  • $\begingroup$ @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 $V_s\neq \{0\}$. $\endgroup$
    – Changyu Guo
    Commented Mar 2, 2015 at 13:09
  • $\begingroup$ @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 $Q=\sum_{i=1}^s i\dim V_i$. $\endgroup$
    – Changyu Guo
    Commented Mar 2, 2015 at 13:14
  • $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Mar 7, 2015 at 10:55

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