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

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

Wedge product

The wedge product of ordinary, real-valued differential forms is defined using multiplication of real numbers. For a pair of Lie algebra–valued differential forms, the wedge product can be defined similarly, but substituting the bilinear Lie bracket operation, to obtain another Lie algebra–valued form. For a [math]\displaystyle{ \mathfrak{g} }[/math]-valued [math]\displaystyle{ p }[/math]-form [math]\displaystyle{ \omega }[/math] and a [math]\displaystyle{ \mathfrak{g} }[/math]-valued [math]\displaystyle{ q }[/math]-form [math]\displaystyle{ \eta }[/math], their wedge product [math]\displaystyle{ [\omega\wedge\eta] }[/math] is given by

[math]\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)})], }[/math]

where the [math]\displaystyle{ v_i }[/math]'s are tangent vectors. The notation is meant to indicate both operations involved. For example, if [math]\displaystyle{ \omega }[/math] and [math]\displaystyle{ \eta }[/math] are Lie-algebra-valued one forms, then one has

[math]\displaystyle{ [\omega\wedge\eta](v_1,v_2) = [\omega(v_1), \eta(v_2)] - [\omega(v_2),\eta(v_1)]. }[/math]

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

[math]\displaystyle{ [(g \otimes \alpha) \wedge (h \otimes \beta)] = [g, h] \otimes (\alpha \wedge \beta) }[/math]

for all [math]\displaystyle{ g, h \in \mathfrak{g} }[/math] and [math]\displaystyle{ \alpha, \beta \in \Omega(M, \mathbb R) }[/math].

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

[math]\displaystyle{ [\omega\wedge\eta] = \omega\wedge\eta - (-1)^{pq}\eta\wedge\omega, }[/math]

where [math]\displaystyle{ \omega \wedge \eta,\ \eta \wedge \omega \in \Omega^{p+q}(M, \mathfrak g) }[/math] are wedge products formed using the matrix multiplication on [math]\displaystyle{ \mathfrak g }[/math].

Operations

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

Similarly, if [math]\displaystyle{ f }[/math] is a multilinear functional on [math]\displaystyle{ \textstyle \prod_1^k \mathfrak{g} }[/math], then one puts

[math]\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)})) }[/math]

where [math]\displaystyle{ q = q_1 + \ldots + q_k }[/math] and [math]\displaystyle{ \varphi_i }[/math] are [math]\displaystyle{ \mathfrak{g} }[/math]-valued [math]\displaystyle{ q_i }[/math]-forms. Moreover, given a vector space [math]\displaystyle{ V }[/math], the same formula can be used to define the [math]\displaystyle{ V }[/math]-valued form [math]\displaystyle{ f(\varphi, \eta) }[/math] when

[math]\displaystyle{ f: \mathfrak{g} \times V \to V }[/math]

is a multilinear map, [math]\displaystyle{ \varphi }[/math] is a [math]\displaystyle{ \mathfrak{g} }[/math]-valued form and [math]\displaystyle{ \eta }[/math] is a [math]\displaystyle{ V }[/math]-valued form. Note that, when

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

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

[math]\displaystyle{ \rho: \mathfrak{g} \to V, \rho(x)y = f(x, y) }[/math]

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

In general, if [math]\displaystyle{ \alpha }[/math] is a [math]\displaystyle{ \mathfrak{gl}(V) }[/math]-valued [math]\displaystyle{ p }[/math]-form and [math]\displaystyle{ \varphi }[/math] is a [math]\displaystyle{ V }[/math]-valued [math]\displaystyle{ q }[/math]-form, then one more commonly writes [math]\displaystyle{ \alpha \cdot \varphi = f(\alpha, \varphi) }[/math] when [math]\displaystyle{ f(T, x) = T x }[/math]. Explicitly,

[math]\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)}). }[/math]

With this notation, one has for example:

[math]\displaystyle{ \operatorname{ad}(\alpha) \cdot \phi = [\alpha \wedge \phi] }[/math].

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

[math]\displaystyle{ \rho([\omega \wedge \omega]) \cdot \varphi = 2 \rho(\omega) \cdot (\rho(\omega) \cdot \varphi). }[/math]

Forms with values in an adjoint bundle

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

[math]\displaystyle{ \mathfrak{g}_P = P \times_{\operatorname{Ad}} \mathfrak{g}. }[/math]

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

See also

• Maurer–Cartan form
• Adjoint bundle

External links

• Wedge Product of Lie Algebra Valued One-Form
• groupoid of Lie-algebra valued forms in nLab