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I am very confused about that in some literature the Maurer Cartan forms for a matrix group is written as

##{\omega _g} = {g^{ - 1}}dg##

what is ##dg## here? can anyone give an example explicitly?

My best guess is

##

dg = \left( {\begin{array}{*{20}{c}}

{d{x^{11}}}& \ldots &{d{x^{1m}}}\\

\vdots & \ddots & \vdots \\

{d{x^{m1}}}& \cdots &{d{x^{mm}}}

\end{array}} \right)% MathType!End!2!1!

##

and if ## V \in {T_e}G##, I can find

##{\left. {{X_V}} \right|_g} = {L_{g * }}V = {\left. {{{(gV)}^{kj}}\frac{\partial }{{\partial {x^{kj}}}}} \right|_g}##

in this way i seem to be able to pullback ##{\left. {{X_V}} \right|_g}##

##\begin{array}{l}

{g^{ - 1}}dg({X_V}) = {({g^{ - 1}})^{ik}}d{x^{kj}}\left( {{{(gV)}^{mn}}\frac{\partial }{{\partial {x^{mn}}}}} \right)\\

= {({g^{ - 1}})^{im}}{(gV)^{mn}} = {({g^{ - 1}}gV)^{in}} = {V^{in}}

\end{array}##

am I right ?

##{\omega _g} = {g^{ - 1}}dg##

what is ##dg## here? can anyone give an example explicitly?

My best guess is

##

dg = \left( {\begin{array}{*{20}{c}}

{d{x^{11}}}& \ldots &{d{x^{1m}}}\\

\vdots & \ddots & \vdots \\

{d{x^{m1}}}& \cdots &{d{x^{mm}}}

\end{array}} \right)% MathType!End!2!1!

##

and if ## V \in {T_e}G##, I can find

##{\left. {{X_V}} \right|_g} = {L_{g * }}V = {\left. {{{(gV)}^{kj}}\frac{\partial }{{\partial {x^{kj}}}}} \right|_g}##

in this way i seem to be able to pullback ##{\left. {{X_V}} \right|_g}##

##\begin{array}{l}

{g^{ - 1}}dg({X_V}) = {({g^{ - 1}})^{ik}}d{x^{kj}}\left( {{{(gV)}^{mn}}\frac{\partial }{{\partial {x^{mn}}}}} \right)\\

= {({g^{ - 1}})^{im}}{(gV)^{mn}} = {({g^{ - 1}}gV)^{in}} = {V^{in}}

\end{array}##

am I right ?

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