The exterior covariant derivative of algebra-valued forms

Recalling the definition of parallel transport of a tensor, we can view a \({gl(n,\mathbb{R})}\)-valued 0-form \({\check{\Theta}}\) as a tensor field of type \({\left(1,1\right)}\), so that the infinitesimal parallel transport of \({\check{\Theta}}\) along \({C}\) with tangent \({v}\) is

\(\displaystyle \parallel_{C}(\check{\Theta})=\left(1-\varepsilon\check{\Gamma}\left(v\right)\right)\check{\Theta}\left(1+\varepsilon\check{\Gamma}\left(v\right)\right). \)

We can now follow the reasoning used to define the covariant derivative of a vector in terms of the connection

\begin{aligned}\nabla_{v}w & \equiv\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left(w\left|_{p+\varepsilon v}\right.-\parallel_{C}\left(w\left|_{p}\right.\right)\right)\\ & =\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left(\vec{w}\left|_{p+\varepsilon v}\right.-\left(1-\varepsilon\check{\Gamma}\left(v\right)\right)\vec{w}\left|_{p}\right.\right)\\ & =\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left(w^{\mu}\left|_{p+\varepsilon v}\right.-w^{\mu}\left|_{p}\right.+\varepsilon\Gamma^{\mu}{}_{\lambda}\left(v\right)w^{\lambda}\left|_{p}\right.\right)e_{\mu}\left|_{p+\varepsilon v}\right.\\ & =\mathrm{d}\vec{w}\left(v\right)+\check{\Gamma}\left(v\right)\vec{w} \end{aligned}

to give the covariant derivative of a \({gl(n,\mathbb{R})}\)-valued 0-form

\begin{aligned}\nabla_{v}\check{\Theta} & \equiv\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left(\check{\Theta}\left|_{p+\varepsilon v}\right.-\parallel_{C}\left(\check{\Theta}\left|_{p}\right.\right)\right)\\ & =\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left(\check{\Theta}\left|_{p+\varepsilon v}\right.-\left(1-\varepsilon\check{\Gamma}\left(v\right)\right)\check{\Theta}\left|_{p}\right.\left(1+\varepsilon\check{\Gamma}\left(v\right)\right)\right)\\ & =\mathrm{d}\check{\Theta}\left(v\right)+\check{\Gamma}\left(v\right)\check{\Theta}-\check{\Theta}\check{\Gamma}\left(v\right)\\ & =\mathrm{d}\check{\Theta}\left(v\right)+\left[\check{\Gamma},\check{\Theta}\right]\left(v\right)\\ & =\mathrm{d}\check{\Theta}\left(v\right)+\left(\check{\Gamma}[\wedge]\check{\Theta}\right)\left(v\right). \end{aligned}

Here we have only kept terms to order \({\varepsilon}\), followed previous convention to define \({\mathrm{d}\check{\Theta}\left(v\right)\equiv\mathrm{d}\Theta^{\mu}{}_{\lambda}\beta^{\lambda}e_{\mu}}\), and defined the Lie commutator \({[\check{\Gamma},\check{\Theta}]}\) in terms of the multiplication of the \({gl(n,\mathbb{R})}\)-valued forms \({\check{\Gamma}}\) and \({\check{\Theta}}\), which recalling our notation from the section on algebra-valued exterior forms as a 1-form is equivalent to \({\check{\Gamma}[\wedge]\check{\Theta}}\). \({\nabla_{v}\check{\Theta}}\) is then “the difference between the linear transformation \({\check{\Theta}}\) and its parallel transport in the direction \({v}\).”

The above definition of the covariant derivative can then be extended to arbitrary \({gl(n,\mathbb{R})}\)-valued \({k}\)-forms by defining

\(\displaystyle \mathrm{D}\check{\Theta}\equiv\mathrm{d}\check{\Theta}+\check{\Gamma}[\wedge]\check{\Theta}, \)

which can be shown to be equivalent to the construction used for \({\mathbb{R}^{n}}\)-valued \({k}\)-forms in the previous section. For example for a \({gl(n,\mathbb{R})}\)-valued 1-form \({\check{\Theta}}\), we have \({\mathrm{D}\check{\Theta}\left(v,w\right)\equiv\nabla_{v}\check{\Theta}\left(w\right)-\nabla_{w}\check{\Theta}\left(v\right)-\check{\Theta}\left(\left[v,w\right]\right)}\), with the covariant derivatives acting on the value of \({\check{\Theta}}\) as a tensor of type \({\left(1,1\right)}\). So at a point \({p}\), \({\mathrm{D}\check{\Theta}\left(v,w\right)}\) can be viewed as the “sum of \({\check{\Theta}}\) on the boundary of the surface defined by its arguments after being parallel transported back to \({p}\).” With respect to the set of \({gl(n,\mathbb{R})}\)-valued forms under the exterior product using the Lie commutator \({[\wedge]}\), \({\mathrm{D}}\) is a graded derivation and for a \({gl(n,\mathbb{R})}\)-valued \({k}\)-form \({\check{\Theta}}\) satisfies the Leibniz rule \({\mathrm{D}(\check{\Theta}[\wedge]\check{\Psi})=\mathrm{D}\check{\Theta}[\wedge]\check{\Psi}+\left(-1\right)^{k}\check{\Theta}[\wedge]\mathrm{D}\check{\Psi}}\).

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