Talk:Equivariant differential form

I think that "invariant element of $\mathbb {C} [{\mathfrak {g}}]\otimes \Omega ^{*}(M).$" shoud be "invariant element of $\mathbb {C} [{\mathfrak {g}}^{*}]\otimes \Omega ^{*}(M).$"? Basically $\operatorname {Mor} ({\mathfrak {g}},\Omega ^{*}(M))=\operatorname {Mor} ({\mathfrak {g}},\mathbb {C} )\otimes \Omega ^{*}(M)=\mathbb {C} [{\mathfrak {g}}^{*}]\otimes \Omega ^{*}(M)$, so I would expect invariant elements to do the same. Of course, the Killing form identifies ${\mathfrak {g}}$ and ${\mathfrak {g}}^{*}$ but it seems cleaner not to invoke the Killing form.

You're correct; "*" is missing (as you can see from your proof as well as one in the article.) -- Taku (talk) 11:14, 7 April 2015 (UTC)

Actually it was correct before; I think it's a matter of the notation. In the article (and elsewhere in Wikipedia), we write $\mathbb {C} [{\mathfrak {g}}]$ for the coordinate ring of ${\mathfrak {g}}$ (see ring of polynomial functions). That is, $\mathbb {C} [{\mathfrak {g}}]=\operatorname {Sym} ({\mathfrak {g}}^{*})$. Since "Sym" also appears in literature, that version is now also noted in the article. -- Taku (talk) 11:36, 7 April 2015 (UTC)