# Derivations

In general, we define a derivation to be a linear map $${\mathcal{D}\colon \mathfrak{a}\to \mathfrak{a}}$$ on an algebra $${\mathfrak{a}}$$ that follows the Leibniz rule (AKA product rule)

$$\displaystyle \mathcal{D}(AB)=(\mathcal{D}A)B+A(\mathcal{D}B).$$

As noted previously, the set $${\mathrm{vect}(M)}$$ of vector fields on a manifold form a Lie algebra; the Lie bracket operation with a fixed vector field $${\left[u,\;\right]}$$ is then a derivation on this algebra, since the Leibniz rule

$$\displaystyle \left[u,\left[v,w\right]\right]=\left[\left[u,v\right],w\right]+\left[v,\left[u,w\right]\right]$$

is just the Jacobi identity.

For a graded algebra, e.g. the exterior algebra, the degree of a derivation is the integer $${c}$$ where $${\mathcal{D}\colon\Lambda^{k}M\to\Lambda^{k+c}M}$$. A graded derivation is defined to follow the graded Liebniz rule, e.g. for a $${k}$$-form $${\varphi}$$,

$$\displaystyle \mathcal{D}\left(\varphi\wedge\psi\right)=\mathcal{D}\varphi\wedge\psi+\left(-1\right)^{kc}\varphi\wedge\mathcal{D}\psi.$$

If $${c}$$ is odd, a graded derivation is sometimes called an anti-derivation (AKA skew-derivation).