# 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 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 $${\mathrm{D}(\check{\Theta}[\wedge]\check{\Psi})=\mathrm{D}\check{\Theta}[\wedge]\check{\Psi}+\left(-1\right)^{k}\check{\Theta}[\wedge]\mathrm{D}\check{\Psi}}$$.