Algebra theory MOC

Derivation on an algebra

A derivation 𝐷 on an algebra (𝐴, ⋅) over a field 𝕂 is a linear endomorphism 𝐷 :𝐴 →𝐴 satisfying the product rule falg

𝐷(𝑎⋅𝑏)=𝐷(𝑎)⋅𝑏+𝑎⋅𝐷(𝑏)

for all 𝑎,𝑏 ∈𝐴. One can more generally define a derivation 𝐷 :𝐴 →𝑀 for any 𝐴-bimodule 𝑀.

Properties

1. The commutator of two derivations is itself a derivation
2. A derivation on a K-monoid is a derivation on its commutator

Proof of 2

Let 𝐴 be a unital associative algebra over 𝕂 and 𝑑 :𝐴 →𝐴 be a derivation of 𝐴. Then for any 𝑎,𝑏 ∈𝐴

𝑑[𝑎,𝑏]=𝑑(𝑎𝑏−𝑏𝑎)=𝑑(𝑎𝑏)−𝑑(𝑏𝑎)=(𝑑𝑎)𝑏+𝑎(𝑑𝑏)−(𝑑𝑏)𝑎−𝑏(𝑑𝑎)=[𝑑𝑎,𝑏]+[𝑎,𝑑𝑏]

proving ^P2.

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