Lie algebra-valued differential form

In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

## Formal definition[edit]

A Lie-algebra-valued differential  k {\displaystyle k} -form on a manifold,  M {\displaystyle M} , is a smooth section of the bundle ( g × M ) ⊗ ∧ k T ∗ M {\displaystyle ({\mathfrak {g}}\times M)\otimes \wedge ^{k}T^{*}M} , where  g {\displaystyle {\mathfrak {g}}} is a Lie algebra,  T ∗ M {\displaystyle T^{*}M} is the cotangent bundle of  M {\displaystyle M} and  ∧ k {\displaystyle \wedge ^{k}} denotes the  k th {\displaystyle k^{\text{th}}} exterior power.

## Wedge product[edit]

Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie-algebra-valued forms can be composed with the bracket operation to obtain another Lie-algebra-valued form. For a  g {\displaystyle {\mathfrak {g}}} -valued  p {\displaystyle p} -form  ω {\displaystyle \omega } and a  g {\displaystyle {\mathfrak {g}}} -valued  q {\displaystyle q} -form  η {\displaystyle \eta } , their wedge product  [ ω ∧ η ] {\displaystyle [\omega \wedge \eta ]} is given by

[ ω ∧ η ] ( v 1 , … , v p + q ) = 1 ( p + q ) ! ∑ σ sgn ⁡ ( σ ) [ ω ( v σ ( 1 ) , … , v σ ( p ) ) , η ( v σ ( p + 1 ) , … , v σ ( p + q ) ) ] , {\displaystyle [\omega \wedge \eta ](v_{1},\dotsc ,v_{p+q})={1 \over (p+q)!}\sum _{\sigma }\operatorname {sgn} (\sigma )[\omega (v_{\sigma (1)},\dotsc ,v_{\sigma (p)}),\eta (v_{\sigma (p+1)},\dotsc ,v_{\sigma (p+q)})],}

where the  v i {\displaystyle v_{i}} 's are tangent vectors. The notation is meant to indicate both operations involved. For example, if  ω {\displaystyle \omega } and  η {\displaystyle \eta } are Lie-algebra-valued one forms, then one has

[ ω ∧ η ] ( v 1 , v 2 ) = 1 2 ( [ ω ( v 1 ) , η ( v 2 ) ] − [ ω ( v 2 ) , η ( v 1 ) ] ) . {\displaystyle [\omega \wedge \eta ](v_{1},v_{2})={1 \over 2}([\omega (v_{1}),\eta (v_{2})]-[\omega (v_{2}),\eta (v_{1})]).}

The operation  [ ω ∧ η ] {\displaystyle [\omega \wedge \eta ]} can also be defined as the bilinear operation on  Ω ( M , g ) {\displaystyle \Omega (M,{\mathfrak {g}})} satisfying

[ ( g ⊗ α ) ∧ ( h ⊗ β ) ] = [ g , h ] ⊗ ( α ∧ β ) {\displaystyle [(g\otimes \alpha )\wedge (h\otimes \beta )]=[g,h]\otimes (\alpha \wedge \beta )}

for all  g , h ∈ g {\displaystyle g,h\in {\mathfrak {g}}} and  α , β ∈ Ω ( M , R ) {\displaystyle \alpha ,\beta \in \Omega (M,\mathbb {R} )} .

Some authors have used the notation  [ ω , η ] {\displaystyle [\omega ,\eta ]} instead of  [ ω ∧ η ] {\displaystyle [\omega \wedge \eta ]} . The notation  [ ω , η ] {\displaystyle [\omega ,\eta ]} , which resembles a commutator, is justified by the fact that if the Lie algebra  g {\displaystyle {\mathfrak {g}}} is a matrix algebra then  [ ω ∧ η ] {\displaystyle [\omega \wedge \eta ]} is nothing but the graded commutator of  ω {\displaystyle \omega } and  η {\displaystyle \eta } , i. e. if  ω ∈ Ω p ( M , g ) {\displaystyle \omega \in \Omega ^{p}(M,{\mathfrak {g}})} and  η ∈ Ω q ( M , g ) {\displaystyle \eta \in \Omega ^{q}(M,{\mathfrak {g}})} then

[ ω ∧ η ] = ω ∧ η − ( − 1 ) p q η ∧ ω , {\displaystyle [\omega \wedge \eta ]=\omega \wedge \eta -(-1)^{pq}\eta \wedge \omega ,}

where  ω ∧ η , η ∧ ω ∈ Ω p + q ( M , g ) {\displaystyle \omega \wedge \eta ,\ \eta \wedge \omega \in \Omega ^{p+q}(M,{\mathfrak {g}})} are wedge products formed using the matrix multiplication on  g {\displaystyle {\mathfrak {g}}} .

## Operations[edit]

Let  f : g -> h {\displaystyle f:{\mathfrak {g}}\to {\mathfrak {h}}} be a Lie algebra homomorphism. If  φ {\displaystyle \varphi } is a  g {\displaystyle {\mathfrak {g}}} -valued form on a manifold, then  f ( φ ) {\displaystyle f(\varphi )} is an  h {\displaystyle {\mathfrak {h}}} -valued form on the same manifold obtained by applying  f {\displaystyle f} to the values of  φ {\displaystyle \varphi } :  f ( φ ) ( v 1 , … , v k ) = f ( φ ( v 1 , … , v k ) ) {\displaystyle f(\varphi )(v_{1},\dotsc ,v_{k})=f(\varphi (v_{1},\dotsc ,v_{k}))} .

Similarly, if  f {\displaystyle f} is a multilinear functional on  ∏ 1 k g {\displaystyle \textstyle \prod _{1}^{k}{\mathfrak {g}}} , then one puts[1]

f ( φ 1 , … , φ k ) ( v 1 , … , v q ) = 1 q ! ∑ σ sgn ⁡ ( σ ) f ( φ 1 ( v σ ( 1 ) , … , v σ ( q 1 ) ) , … , φ k ( v σ ( q − q k + 1 ) , … , v σ ( q ) ) ) {\displaystyle f(\varphi _{1},\dotsc ,\varphi _{k})(v_{1},\dotsc ,v_{q})={1 \over q!}\sum _{\sigma }\operatorname {sgn} (\sigma )f(\varphi _{1}(v_{\sigma (1)},\dotsc ,v_{\sigma (q_{1})}),\dotsc ,\varphi _{k}(v_{\sigma (q-q_{k}+1)},\dotsc ,v_{\sigma (q)}))}

where  q = q 1 + … + q k {\displaystyle q=q_{1}+\ldots +q_{k}} and  φ i {\displaystyle \varphi _{i}} are  g {\displaystyle {\mathfrak {g}}} -valued  q i {\displaystyle q_{i}} -forms. Moreover, given a vector space  V {\displaystyle V} , the same formula can be used to define the  V {\displaystyle V} -valued form  f ( φ , η ) {\displaystyle f(\varphi ,\eta )} when

f : g × V -> V {\displaystyle f:{\mathfrak {g}}\times V\to V}

is a multilinear map,  φ {\displaystyle \varphi } is a  g {\displaystyle {\mathfrak {g}}} -valued form and  η {\displaystyle \eta } is a  V {\displaystyle V} -valued form. Note that, when

f ( [ x , y ] , z ) = f ( x , f ( y , z ) ) − f ( y , f ( x , z ) ) , ( ∗ ) {\displaystyle f([x,y],z)=f(x,f(y,z))-f(y,f(x,z)){,}\qquad (*)}

giving  f {\displaystyle f} amounts to giving an action of  g {\displaystyle {\mathfrak {g}}} on  V {\displaystyle V} ; i.e.,  f {\displaystyle f} determines the representation

ρ : g -> V , ρ ( x ) y = f ( x , y ) {\displaystyle \rho :{\mathfrak {g}}\to V,\rho (x)y=f(x,y)}

and, conversely, any representation  ρ {\displaystyle \rho } determines  f {\displaystyle f} with the condition  ( ∗ ) {\displaystyle (*)} . For example, if  f ( x , y ) = [ x , y ] {\displaystyle f(x,y)=[x,y]} (the bracket of  g {\displaystyle {\mathfrak {g}}} ), then we recover the definition of  [ ⋅ ∧ ⋅ ] {\displaystyle [\cdot \wedge \cdot ]} given above, with  ρ = ad {\displaystyle \rho =\operatorname {ad} } , the adjoint representation. (Note the relation between  f {\displaystyle f} and  ρ {\displaystyle \rho } above is thus like the relation between a bracket and  ad {\displaystyle \operatorname {ad} } .)

In general, if  α {\displaystyle \alpha } is a  g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} -valued  p {\displaystyle p} -form and  φ {\displaystyle \varphi } is a  V {\displaystyle V} -valued  q {\displaystyle q} -form, then one more commonly writes  α ⋅ φ = f ( α , φ ) {\displaystyle \alpha \cdot \varphi =f(\alpha ,\varphi )} when  f ( T , x ) = T x {\displaystyle f(T,x)=Tx} . Explicitly,

( α ⋅ ϕ ) ( v 1 , … , v p + q ) = 1 ( p + q ) ! ∑ σ sgn ⁡ ( σ ) α ( v σ ( 1 ) , … , v σ ( p ) ) ϕ ( v σ ( p + 1 ) , … , v σ ( p + q ) ) . {\displaystyle (\alpha \cdot \phi )(v_{1},\dotsc ,v_{p+q})={1 \over (p+q)!}\sum _{\sigma }\operatorname {sgn} (\sigma )\alpha (v_{\sigma (1)},\dotsc ,v_{\sigma (p)})\phi (v_{\sigma (p+1)},\dotsc ,v_{\sigma (p+q)}).}

With this notation, one has for example:

ad ⁡ ( α ) ⋅ ϕ = [ α ∧ ϕ ] {\displaystyle \operatorname {ad} (\alpha )\cdot \phi =[\alpha \wedge \phi ]} .

Example: If  ω {\displaystyle \omega } is a  g {\displaystyle {\mathfrak {g}}} -valued one-form (for example, a connection form),  ρ {\displaystyle \rho } a representation of  g {\displaystyle {\mathfrak {g}}} on a vector space  V {\displaystyle V} and  φ {\displaystyle \varphi } a  V {\displaystyle V} -valued zero-form, then

ρ ( [ ω ∧ ω ] ) ⋅ φ = 2 ρ ( ω ) ⋅ ( ρ ( ω ) ⋅ φ ) . {\displaystyle \rho ([\omega \wedge \omega ])\cdot \varphi =2\rho (\omega )\cdot (\rho (\omega )\cdot \varphi ).} [2]

## Forms with values in an adjoint bundle[edit]

See also: adjoint bundle

Let  P {\displaystyle P} be a smooth principal bundle with structure group  G {\displaystyle G} and  g = Lie ⁡ ( G ) {\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)} .  G {\displaystyle G} acts on  g {\displaystyle {\mathfrak {g}}} via adjoint representation and so one can form the associated bundle:

g P = P × Ad g . {\displaystyle {\mathfrak {g}}_{P}=P\times _{\operatorname {Ad} }{\mathfrak {g}}.}

Any  g P {\displaystyle {\mathfrak {g}}_{P}} -valued forms on the base space of  P {\displaystyle P} are in a natural one-to-one correspondence with any tensorial forms on  P {\displaystyle P} of adjoint type.

## See also[edit]

* Maurer–Cartan form
* Adjoint bundle

## Notes[edit]

1. ^ Kobayashi–Nomizu, Ch. XII, § 1. harvnb error: no target: CITEREFKobayashi–Nomizu (help)
2. ^ Since  ρ ( [ ω ∧ ω ] ) ( v , w ) = ρ ( [ ω ∧ ω ] ( v , w ) ) = ρ ( [ ω ( v ) , ω ( w ) ] ) = ρ ( ω ( v ) ) ρ ( ω ( w ) ) − ρ ( ω ( w ) ) ρ ( ω ( v ) ) {\displaystyle \rho ([\omega \wedge \omega ])(v,w)=\rho ([\omega \wedge \omega ](v,w))=\rho ([\omega (v),\omega (w)])=\rho (\omega (v))\rho (\omega (w))-\rho (\omega (w))\rho (\omega (v))} , we have that

( ρ ( [ ω ∧ ω ] ) ⋅ φ ) ( v , w ) = 1 2 ( ρ ( [ ω ∧ ω ] ) ( v , w ) φ − ρ ( [ ω ∧ ω ] ) ( w , v ) ϕ ) {\displaystyle (\rho ([\omega \wedge \omega ])\cdot \varphi )(v,w)={1 \over 2}(\rho ([\omega \wedge \omega ])(v,w)\varphi -\rho ([\omega \wedge \omega ])(w,v)\phi )}
is  ρ ( ω ( v ) ) ρ ( ω ( w ) ) φ − ρ ( ω ( w ) ) ρ ( ω ( v ) ) ϕ = 2 ( ρ ( ω ) ⋅ ( ρ ( ω ) ⋅ ϕ ) ) ( v , w ) . {\displaystyle \rho (\omega (v))\rho (\omega (w))\varphi -\rho (\omega (w))\rho (\omega (v))\phi =2(\rho (\omega )\cdot (\rho (\omega )\cdot \phi ))(v,w).}

## References[edit]

* S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2.

## External links[edit]

* Wedge Product of Lie Algebra Valued One-Form
* groupoid of Lie-algebra valued forms at the nLab