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 $$k$$-form on a manifold, $$M$$, is a smooth section of the bundle $$ (\mathfrak{g} \times M) \otimes \wedge^k T^*M$$, where $$\mathfrak{g}$$ is a Lie algebra, $$T^*M$$ is the cotangent bundle of $$M$$ and $$\wedge^k$$ denotes the $$k^{\text{th}}$$ 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 $$\mathfrak{g}$$-valued $$p$$-form $$\omega$$ and a $$\mathfrak{g}$$-valued $$q$$-form $$\eta$$, their wedge product $$[\omega\wedge\eta]$$ is given by
 * $$[\omega\wedge\eta](v_1, \dotsc, v_{p+q}) = {1 \over p!q!}\sum_{\sigma} \operatorname{sgn}(\sigma) [\omega(v_{\sigma(1)}, \dotsc, v_{\sigma(p)}), \eta(v_{\sigma(p+1)}, \dotsc, v_{\sigma(p+q)})],$$

where the $$v_i$$'s are tangent vectors. The notation is meant to indicate both operations involved. For example, if $$\omega$$ and $$\eta$$ are Lie-algebra-valued one forms, then one has
 * $$[\omega\wedge\eta](v_1,v_2) = [\omega(v_1), \eta(v_2)] - [\omega(v_2),\eta(v_1)].$$

The operation $$[\omega\wedge\eta]$$ can also be defined as the bilinear operation on $$\Omega(M, \mathfrak{g})$$ satisfying
 * $$[(g \otimes \alpha) \wedge (h \otimes \beta)] = [g, h] \otimes (\alpha \wedge \beta)$$

for all $$g, h \in \mathfrak{g}$$ and $$\alpha, \beta \in \Omega(M, \mathbb R)$$.

Some authors have used the notation $$[\omega, \eta]$$ instead of $$[\omega\wedge\eta]$$. The notation $$[\omega, \eta]$$, which resembles a commutator, is justified by the fact that if the Lie algebra $$\mathfrak g$$ is a matrix algebra then $$[\omega\wedge\eta]$$ is nothing but the graded commutator of $$\omega$$ and $$\eta$$, i. e. if $$\omega \in \Omega^p(M, \mathfrak g)$$ and $$\eta \in \Omega^q(M, \mathfrak g)$$ then
 * $$[\omega\wedge\eta] = \omega\wedge\eta - (-1)^{pq}\eta\wedge\omega,$$

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

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

Similarly, if $$f$$ is a multilinear functional on $$\textstyle \prod_1^k \mathfrak{g}$$, then one puts
 * $$f(\varphi_1, \dotsc, \varphi_k)(v_1, \dotsc, v_q) = {1 \over q!} \sum_{\sigma} \operatorname{sgn}(\sigma) f(\varphi_1(v_{\sigma(1)}, \dotsc, v_{\sigma(q_1)}), \dotsc, \varphi_k(v_{\sigma(q - q_k + 1)}, \dotsc, v_{\sigma(q)}))$$

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

is a multilinear map, $$\varphi$$ is a $$\mathfrak{g}$$-valued form and $$\eta$$ is a $$V$$-valued form. Note that, when
 * $$f([x, y], z) = f(x, f(y, z)) - f(y, f(x, z)) {,} \qquad (*)$$

giving $$f$$ amounts to giving an action of $$\mathfrak{g}$$ on $$V$$; i.e., $$f$$ determines the representation
 * $$\rho: \mathfrak{g} \to V, \rho(x)y = f(x, y)$$

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

In general, if $$\alpha$$ is a $$\mathfrak{gl}(V)$$-valued $$p$$-form and $$\varphi$$ is a $$V$$-valued $$q$$-form, then one more commonly writes $$\alpha \cdot \varphi = f(\alpha, \varphi)$$ when $$f(T, x) = T x$$. Explicitly,
 * $$(\alpha \cdot \phi)(v_1, \dotsc, v_{p+q}) = {1 \over (p+q)!} \sum_{\sigma} \operatorname{sgn}(\sigma) \alpha(v_{\sigma(1)}, \dotsc, v_{\sigma(p)}) \phi(v_{\sigma(p+1)}, \dotsc, v_{\sigma(p+q)}).$$

With this notation, one has for example:
 * $$\operatorname{ad}(\alpha) \cdot \phi = [\alpha \wedge \phi]$$.

Example: If $$\omega$$ is a $$\mathfrak{g}$$-valued one-form (for example, a connection form), $$\rho$$ a representation of $$\mathfrak{g}$$ on a vector space $$V$$ and $$\varphi$$ a $$V$$-valued zero-form, then
 * $$\rho([\omega \wedge \omega]) \cdot \varphi = 2 \rho(\omega) \cdot (\rho(\omega) \cdot \varphi).$$

Forms with values in an adjoint bundle
Let $$P$$ be a smooth principal bundle with structure group $$G$$ and $$\mathfrak{g} = \operatorname{Lie}(G)$$. $$G$$ acts on $$\mathfrak{g}$$ via adjoint representation and so one can form the associated bundle:
 * $$\mathfrak{g}_P = P \times_{\operatorname{Ad}} \mathfrak{g}.$$

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