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As i am sitting here reading some lecture notes on lie algebras I found myself getting stock because of the word "rank". As I understand rank, it's just the dimension of the image of a linear map and therefore rank is just a fixed number associated to a particular linear map. I have tagged a part of the notes, and as you can see there are several possible rank's for a certain exterior derivative say $d: \wedge^1 g^* \rightarrow \wedge^2 g^* $. The notes goes on to rank 2 and 3 as well. I'm sure this is just me missing something obvious, but I can't figure out how rank is being defined in this context.

Can someone help me understand this please?

notes on lie algebras

Martinius
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  • I'm not sure but maybe your confusion comes from a bad notation from the author, as he named $d$ both the application $\wedge$ definied over $g^{\star}$ and the application $\wedge$ definied over $\wedge^{1} g^{\star}$. – krirkrirk Nov 01 '15 at 11:43
  • Let me clarify my confusion.. Since we are dealing with Lie algebras of dimension 3 then dim$\wedge^2 g^* = 3$, and so I would think that the rank of $d: \wedge^1 g^* \rightarrow \wedge^2 g^* $ would be 3. So how can the rank be either 0,1,2 or 3? – Martinius Nov 01 '15 at 12:21
  • @Martinius: the rank of a linear map $f : V \to W$ is not the dimension of $W$. It's the dimension of the image $\text{im}(f) = { f(v) : v \in V }$, which can be any number from $0$ to $\dim W$. – Qiaochu Yuan Nov 01 '15 at 18:17
  • Of course yes:) Thank you for clarifying the obvious. – Martinius Nov 01 '15 at 18:25

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