nLab derivation Lie algebra

Definition

For $\mathfrak{g}$ a Lie algebra, then the Lie algebra of its automorphism Lie group

$$\mathfrak{aut}(\mathfrak{g})_{\mathrm{even}}$$

called the the automorphism Lie algebra of $\mathfrak{g}$ (or derivation Lie algebra), is the Lie algebra whose underlying vector space is that of those linear maps $\Delta \colon \mathfrak{g} \to \mathfrak{g}$ which satisfy the derivation property:

$$\Delta([x,y]) = [\Delta(x), y] + [x, \Delta(y)]$$

for all $x,y \in \mathfrak{g}$. The Lie bracket on $\mathfrak{aut}(\mathfrak{g})_{\mathrm{even}}$ is the commutator operation:

$$[\Delta_1, \Delta_2] := \Delta_1 \circ \Delta_2 - \Delta_2 \circ \Delta_1 \,.$$

Related concepts

• endomorphism Lie algebra

• automorphism infinity-Lie algebra