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derivation Lie algebra

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Definition

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

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

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

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

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

[Δ 1,Δ 2]:=Δ 1Δ 2Δ 2Δ 1. [\Delta_1, \Delta_2] := \Delta_1 \circ \Delta_2 - \Delta_2 \circ \Delta_1 \,.

Created on March 7, 2017 at 04:48:54. See the history of this page for a list of all contributions to it.