Maurer-Cartan form

Categories: Lie groups | Equations

In mathematics, the Maurer-Cartan form for a Lie group G is a distinguished differential form on G that carries within itself the basic infinitesimal information about the structure of G. It was much used by Elie Cartan, as a basic ingredient of his method of moving frames.

Let <math>g=T_eG</math> be the tangent space of a Lie group G at the identity (its Lie algebra). G acts on itself by left translation

<math>L_h:G\ni k\mapsto hk\in G</math>,

and this induces a map of the tangent bundle to itself

<math>(L_h)_*:T_kG\rightarrow T_{hk}G</math>.

A left-invariant vector field is a section X of <math>TG</math> such that

The Maurer-Cartan form <math>\omega</math> is a g-valued one-form on G defined on vectors <math>v\in T_h G </math> by the formula <math>\omega(v)=(L_{h^{-1}})_*v\in g</math>. If X is a left-invariant vector field on G, then <math>\omega(X)</math> is constant on G. Furthermore, if X and Y are both left-invariant, then

<math>\omega([X,Y])=[\omega(X),\omega(Y)]</math>

where the bracket on the LHS is the Lie bracket of vector fields, and the bracket on the RHS is the bracket on the Lie algebra g. (This may be used as the definition of the bracket on g.) These facts may be used to establish an isomorphism of Lie algebras

<math>g=T_eG\cong \{\hbox{left-invariant vector fields on G}\}</math>.

By the definition of the differential (mathematics), if X and Y are arbitrary vector fields then

<math> d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y])</math>.

In particular, if X and Y are left-invariant, then

<math>X(\omega(Y))=Y(\omega(X))=0</math>,

<math>d\omega(X,Y)+[\omega(X),\omega(Y)]=0</math>

but the left-hand side is simply a 2-form, so the equation does not rely on the fact that X and Y are left-invariant. The conclusion follows that the equation is true for any pair of vector fields X and Y. This is known as the Maurer-Cartan equation.

If G is embedded in GL(n), then one can write explicit formula for <math> \omega </math>

<math> \omega = g^{-1} dg </math>

If we introduce principal bundle over the Lie group G with transition functions defined by the left action on G, then the connection form <math> A =\omega </math> will be flat. Indeed

<math> F=dA + A \wedge A = 0</math>

is the same as Maurer-Cartan equationzh:Maurer-Cartan形式