Algebra-valued exterior forms

We can extend the view of exterior forms as real-valued linear mappings to define algebra-valued forms. These follow the same construction as in the prior section, starting from an algebra-valued 1-form \({\check{\Theta}\colon V\rightarrow \mathfrak{a}}\), so that general forms are alternating multilinear maps from \({k}\) vectors to a real algebra \({\mathfrak{a}}\) whose vector multiplication takes the place of multiplication in \({\mathbb{R}}\). Since this vector multiplication may not be commutative, we need to be more careful in terms of ordering in the isomorphism to ensure antisymmetry, i.e. for two algebra-valued 1-forms we define

\begin{aligned} (\check{\Theta}\wedge\check{\Psi})(v,w)\equiv\check{\Theta}(v)\check{\Psi}(w)-\check{\Theta}(w)\check{\Psi}(v). \end{aligned}

An algebra-valued form whose values are defined by matrices is a matrix-valued form. Exterior forms that take values in a matrix group can also be considered as matrix-valued forms, but it must be understood that under addition the values may no longer be in the group. One can also form the exterior product between a matrix-valued form and a vector-valued form. To reduce confusion when dealing with algebra- and vector-valued forms, we will indicate them with (non-standard) decorations, for example in the case of a matrix-valued 1-form acting on a vector-valued 1-form,

\begin{aligned} (\check{\Theta}\wedge\vec{\varphi})(v,w)\equiv\check{\Theta}(v)\vec{\varphi}(w)-\check{\Theta}(w)\vec{\varphi}(v). \end{aligned}

Δ An additional distinction can be made between forms that take values which are concrete matrices and column vectors (and thus depend upon the basis of the underlying vector space), and forms that take values which are abstract linear transformations and abstract vectors (and thus are basis-independent). We will attempt to distinguish between these by referring to the specific matrix or abstract group, and by only using “vector-valued” when the value is an abstract vector.

A notational issue arises in the particular case of Lie algebra valued forms, where the related associative algebra in the relation \({[\check{\Theta},\check{\Psi}]=\check{\Theta}\check{\Psi}-\check{\Psi}\check{\Theta}}\) could also be in use. In this case multiplication of values could use either the Lie commutator or that of the related associative algebra. We will denote the exterior product using the Lie commutator by \({\check{\Theta}[\wedge]\check{\Psi}}\). Some authors use \({[\check{\Theta},\check{\Psi}]}\) or \({[\check{\Theta}\wedge\check{\Psi}]}\), but both can be ambiguous, motivating us to introduce our (non-standard) notation. The expression \({\check{\Theta}\wedge\check{\Psi}}\) is then reserved for the exterior product using the underlying associative algebra (e.g. that of matrix multiplication if the associative algebra is defined this way). For two Lie algebra-valued 1-forms we then have

\begin{aligned} (\check{\Theta}[\wedge]\check{\Psi})(v,w) & =[\check{\Theta}(v),\check{\Psi}(w)]-[\check{\Theta}(w),\check{\Psi}(v)]\\ & =\check{\Theta}(v)\check{\Psi}(w)-\check{\Psi}(w)\check{\Theta}(v)-\check{\Theta}(w)\check{\Psi}(v)+\check{\Psi}(v)\check{\Theta}(w). \end{aligned}

Note that \({[\check{\Theta},\check{\Psi}](v,w)=\check{\Theta}(v)\check{\Psi}(w)-\check{\Psi}(v)\check{\Theta}(w)}\) is a distinct construction, as is \({[\check{\Theta}(v),\check{\Psi}(w)]=\check{\Theta}(v)\check{\Psi}(w)-\check{\Psi}(w)\check{\Theta}(v)}\); neither are in general anti-symmetric and thus do not yield forms. Also note that e.g. for two 1-forms \({\check{\Theta}[\wedge]\check{\Psi}\neq\check{\Theta}\wedge\check{\Psi}-\check{\Psi}\wedge\check{\Theta}}\), and \({(\check{\Theta}[\wedge]\check{\Theta})(v,w)=2[\check{\Theta}(v),\check{\Theta}(w)]}\) does not in general vanish, so \({[\wedge]}\) does not act like a Lie commutator in these respects. Instead it forms a graded Lie algebra, so that for algebra-valued \({j}\)- and \({k}\)-forms \({\check{\Theta}}\) and \({\check{\Psi}}\) we have the graded commutativity rule

\(\displaystyle \check{\Theta}[\wedge]\check{\Psi}=(-1)^{jk+1}\check{\Psi}[\wedge]\check{\Theta}, \)

and with an algebra-valued \({m}\)-form \({\check{\Xi}}\) we have the graded Jacobi identity

\(\displaystyle (-1)^{jm}(\check{\Theta}[\wedge]\check{\Psi})[\wedge]\check{\Xi}+(-1)^{kj}(\check{\Psi}[\wedge]\check{\Xi})[\wedge]\check{\Theta}+(-1)^{mk}(\check{\Xi}[\wedge]\check{\Theta})[\wedge]\check{\Psi}=0. \)

Algebra-valued forms also introduce potentially ambiguous index notation. If a basis is chosen for the algebra \({\mathfrak{a}}\), the value of an algebra-valued form may be expressed using component notation as \({\Theta^{\mu}}\); or if the algebra is defined in terms of matrices, an element might be written \({\Theta^{\alpha}{}_{\beta}}\), an expression that has nothing to do with the basis of \({\mathfrak{a}}\). Then for example an algebra-valued 1-form might be written \({\Theta^{\mu}{}_{\gamma}}\) or \({\Theta^{\alpha}{}_{\beta\gamma}}\).

Δ In considering algebra-valued forms expressed in index notation, extra care must be taken to identify the type of form in question, and to match each index with the aspect of the object it was meant to represent.

An Illustrated Handbook