View source for Lie algebra-valued differential form

In differential geometry, a '''Lie-algebra-valued form''' is a [[Vector-valued differential form|differential form with values in a Lie algebra]]. Such forms have important applications in the theory of [[Connection (principal bundle)|connections]] on a [[Principal bundle|principal bundle]] as well as in the theory of [[Cartan connection]]s.
== Formal definition ==

A Lie-algebra-valued differential <math>k</math>-form on a manifold, <math>M</math>, is a smooth [[Section (fiber bundle)|section]] of the bundle <math> (\mathfrak{g} \times M) \otimes \wedge^k T^*M</math>, where <math>\mathfrak{g}</math> is a [[Lie algebra]], <math>T^*M</math> is the [[Cotangent bundle|cotangent bundle]] of <math>M</math> and <math>\wedge^k</math> denotes the <math>k^{\text{th}}</math> [[Exterior algebra|exterior power]].

== Wedge product ==

Since every Lie algebra has a bilinear [[Lie algebra#Definition of a Lie algebra|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 <math>\mathfrak{g}</math>-valued <math>p</math>-form <math>\omega</math> and a <math>\mathfrak{g}</math>-valued <math>q</math>-form <math>\eta</math>, their wedge product <math>[\omega\wedge\eta]</math> is given by 
:<math>[\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)})],</math>
where the <math>v_i</math>'s are tangent vectors. The notation is meant to indicate both operations involved. For example, if <math>\omega</math> and <math>\eta</math> are Lie-algebra-valued one forms, then one has
:<math>[\omega\wedge\eta](v_1,v_2) = {1 \over 2} ([\omega(v_1), \eta(v_2)] - [\omega(v_2),\eta(v_1)]).</math>
<!-- With this operation the set of all Lie-algebra-valued forms on a manifold ''M'' becomes a graded Lie superalgebra.-->

The operation <math>[\omega\wedge\eta]</math> can also be defined as the bilinear operation on <math>\Omega(M, \mathfrak{g})</math> satisfying
:<math>[(g \otimes \alpha) \wedge (h \otimes \beta)] = [g, h] \otimes (\alpha \wedge \beta)</math>
for all <math>g, h \in \mathfrak{g}</math> and <math>\alpha, \beta \in \Omega(M, \mathbb R)</math>.

Some authors have used the notation <math>[\omega, \eta]</math> instead of <math>[\omega\wedge\eta]</math>. The notation <math>[\omega, \eta]</math>, which resembles a [[Commutator#Ring theory|commutator]], is justified by the fact that if the Lie algebra <math>\mathfrak g</math> is a matrix algebra then <math>[\omega\wedge\eta]</math> is nothing but the graded commutator of <math>\omega</math> and <math>\eta</math>, i. e. if <math>\omega \in \Omega^p(M, \mathfrak g)</math> and <math>\eta \in \Omega^q(M, \mathfrak g)</math> then
:<math>[\omega\wedge\eta] = \omega\wedge\eta - (-1)^{pq}\eta\wedge\omega,</math>
where <math>\omega \wedge \eta,\ \eta \wedge \omega \in \Omega^{p+q}(M, \mathfrak g)</math> are wedge products formed using the matrix multiplication on <math>\mathfrak g</math>.

== Operations ==
Let <math>f : \mathfrak{g} \to \mathfrak{h}</math> be a Lie algebra homomorphism. If <math>\varphi</math> is a <math>\mathfrak{g}</math>-valued form on a manifold, then <math>f(\varphi)</math> is an <math>\mathfrak{h}</math>-valued form on the same manifold obtained by applying <math>f</math> to the values of <math>\varphi</math>: <math>f(\varphi)(v_1, \dotsc, v_k) = f(\varphi(v_1, \dotsc, v_k))</math>.

Similarly, if <math>f</math> is a multilinear functional on <math>\textstyle \prod_1^k \mathfrak{g}</math>, then one puts<ref>{{harvnb|Kobayashi–Nomizu|loc=Ch. XII, § 1.}}</ref>
:<math>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)}))</math>
where <math>q = q_1 + \ldots + q_k</math> and <math>\varphi_i</math> are <math>\mathfrak{g}</math>-valued <math>q_i</math>-forms. Moreover, given a vector space <math>V</math>, the same formula can be used to define the <math>V</math>-valued form <math>f(\varphi, \eta)</math> when
:<math>f: \mathfrak{g} \times V \to V</math>
is a multilinear map, <math>\varphi</math> is a <math>\mathfrak{g}</math>-valued form and <math>\eta</math> is a <math>V</math>-valued form. Note that, when
:<math>f([x, y], z) = f(x, f(y, z)) - f(y, f(x, z)) {,} \qquad (*)</math>
giving <math>f</math> amounts to giving an action of <math>\mathfrak{g}</math> on <math>V</math>; i.e., <math>f</math> determines the representation
:<math>\rho: \mathfrak{g} \to V, \rho(x)y = f(x, y)</math>
and, conversely, any representation <math>\rho</math> determines <math>f</math> with the condition <math>(*)</math>. For example, if <math>f(x, y) = [x, y]</math> (the bracket of <math>\mathfrak{g}</math>), then we recover the definition of <math>[\cdot \wedge \cdot]</math> given above, with <math>\rho = \operatorname{ad}</math>, the adjoint representation. (Note the relation between <math>f</math> and <math>\rho</math> above is thus like the relation between a bracket and <math>\operatorname{ad}</math>.) 

In general, if <math>\alpha</math> is a <math>\mathfrak{gl}(V)</math>-valued <math>p</math>-form and <math>\varphi</math> is a <math>V</math>-valued <math>q</math>-form, then one more commonly writes <math>\alpha \cdot \varphi = f(\alpha, \varphi)</math> when <math>f(T, x) = T x</math>. Explicitly,
:<math>(\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)}).</math>
With this notation, one has for example:
:<math>\operatorname{ad}(\alpha) \cdot \phi = [\alpha \wedge \phi]</math>.

Example: If <math>\omega</math> is a <math>\mathfrak{g}</math>-valued one-form (for example, a [[Connection form|connection form]]), <math>\rho</math> a representation of <math>\mathfrak{g}</math> on a vector space <math>V</math> and <math>\varphi</math> a <math>V</math>-valued zero-form, then
:<math>\rho([\omega \wedge \omega]) \cdot \varphi = 2 \rho(\omega) \cdot (\rho(\omega) \cdot \varphi).</math><ref>Since <math>\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))</math>, we have that
:<math>(\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)</math>
is <math>\rho(\omega(v))\rho(\omega(w))\varphi - \rho(\omega(w))\rho(\omega(v))\phi = 2(\rho(\omega) \cdot (\rho(\omega) \cdot \phi))(v, w).</math></ref>

== Forms with values in an adjoint bundle ==
Let <math>P</math> be a smooth principal bundle with structure group <math>G</math> and <math>\mathfrak{g} = \operatorname{Lie}(G)</math>. <math>G</math> acts on <math>\mathfrak{g}</math> via [[Adjoint representation|adjoint representation]] and so one can form the associated bundle:
:<math>\mathfrak{g}_P = P \times_{\operatorname{Ad}} \mathfrak{g}.</math>
Any <math>\mathfrak{g}_P</math>-valued forms on the base space of <math>P</math> are in a natural one-to-one correspondence with any tensorial forms on <math>P</math> of adjoint type.

== See also ==
*[[Maurer–Cartan form]]
*[[Adjoint bundle]]

== Notes ==
{{Reflist}}

== References ==

== External links ==
*[https://mathoverflow.net/q/123632 Wedge Product of Lie Algebra Valued One-Form]
* {{nlab|id=groupoid+of+Lie-algebra+valued+forms|title=groupoid of Lie-algebra valued forms}}

[[Category:Differential forms]]
[[Category:Lie algebras]]

{{Sourceattribution|Lie algebra-valued differential form|1}}