Mason Cartan Shape

The Maurer-Cartan form is a Lie algebra valued differential form on Lie groups that is frequently used in differential geometry and mathematical physics . It is named after the German mathematician and university professor Ludwig Maurer and the French mathematician Élie Cartan .

definition

Be a Lie group, its Lie algebra . For induces the left multiplication ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}=T_{e}G}$ ${\displaystyle g\in G}$

${\displaystyle L_{g^{-1}}:G\rightarrow G}$

${\displaystyle L_{g^{-1}}(h):=g^{-1}h}$

the differential

${\displaystyle (DL_{g^{-1}})_{g}:T_{g}G\rightarrow T_{e}G={\mathfrak {g}}}$.

The mason cartan shape is defined by ${\displaystyle \omega \in \Omega ^{1}(G,{\mathfrak {g}})}$

${\displaystyle \omega (v):=(DL_{g^{-1}})_{g}(v)}$

for . ${\displaystyle v\in T_{g}G,g\in G}$

Mason-Cartan equation

The Maurer-Cartan form satisfies the equation

${\displaystyle d\omega +{\frac {1}{2}}\left[\omega ,\omega \right]=0}$.

Here the commutator is Lie algebra valued differential forms by

${\displaystyle [\omega \wedge \eta ](v_{1},v_{2})=[\omega (v_{1}),\eta (v_{2})]-[\omega (v_{2}),\eta (v_{1})]}$

and the outer derivative through ${\displaystyle d\omega }$

${\displaystyle d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y])}$

Are defined.

Individual evidence

1. ↑ Jeffrey M. Lee, Manifolds and differential geometry . American Mathematical Society, Providence, RI 2009, ISBN 0-8218-4815-1 , Chapter: 5.6 The Maurer Cartan Form.