Derivation on an algebra

A derivation $𝐷$ on an algebra $(𝐴, \cdot)$ over a field $𝕂$ is a linear endomorphism $𝐷 : 𝐴 \to 𝐴$ satisfying the product rule #m/def/ralg

𝐷 (𝑎 \cdot 𝑏) = 𝐷 (𝑎) \cdot 𝑏 + 𝑎 \cdot 𝐷 (𝑏)

for all $𝑎, 𝑏 \in 𝐴$ . One can more generally define a derivation $𝐷 : 𝐴 \to 𝑀$ for any $𝐴$ -bimodule $𝑀$ .

Properties

Proof of 2

Let $𝐴$ be a unital associative algebra over $𝕂$ and $𝑑 : 𝐴 \to 𝐴$ be a derivation of $𝐴$ . Then for any $𝑎, 𝑏 \in 𝐴$

𝑑 [𝑎, 𝑏] = 𝑑 (𝑎 𝑏 - 𝑏 𝑎) = 𝑑 (𝑎 𝑏) - 𝑑 (𝑏 𝑎) = (𝑑 𝑎) 𝑏 + 𝑎 (𝑑 𝑏) - (𝑑 𝑏) 𝑎 - 𝑏 (𝑑 𝑎) = [𝑑 𝑎, 𝑏] + [𝑎, 𝑑 𝑏]

proving ^P2.

#state/tidy | #lang/en | #SemBr