Relationships between derivations

We can define one other derivation on \({k}\)-forms, the interior derivative (AKA inner derivative, inner multiplication), which is the generalization of the interior product to forms on manifolds; i.e. for a given vector \({v}\) it is the graded degree \({-1}\) derivation \({\left(i_{v}\varphi\right)\left(w_{2},\dotsc,w_{k}\right)\equiv\varphi\left(v,w_{2},\dotsc,w_{k}\right)}\) on \({k}\)-forms \({\varphi}\), which follows the graded Leibniz rule \({i_{v}\left(\varphi\wedge\psi\right)=(i_{v}\varphi)\wedge\psi+\left(-1\right)^{k}\varphi\wedge(i_{v}\psi)}\). The graded commutativity of forms immediately gives the property \({i_{v}i_{w}+i_{w}i_{v}=i_{v}^{2}=0}\).

The interior, exterior, and Lie derivatives then form an infinite-dimensional graded Lie algebra with the following relations:

  • \({\left[L_{v},L_{w}\right]\equiv L_{v}L_{w}-L_{w}L_{v}=L_{\left[v,w\right]}}\)
  • \({\left[i_{v},i_{w}\right]\equiv i_{v}i_{w}+i_{w}i_{v}=0}\)
  • \({\left[\mathrm{d},\mathrm{d}\right]\equiv \mathrm{d}^{2}+\mathrm{d}^{2}=0}\)
  • \({\left[L_{v},i_{w}\right]\equiv L_{v}i_{w}-i_{w}L_{v}=i_{\left[v,w\right]}}\)
  • \({\left[L_{v},\mathrm{d}\right]\equiv L_{v}\mathrm{d}-\mathrm{d}L_{v}=0}\)
  • \({\left[i_{v},\mathrm{d}\right]\equiv i_{v}\mathrm{d}+\mathrm{d}i_{v}=L_{v}}\)

An Illustrated Handbook