Tensor product of a Lie algebra and a commutative algebra

Let $𝔤$ be a Lie algebra and $𝐴$ be a commutative algebra, both over $𝕂$ . Then their tensor product algebra $𝔤 \otimes 𝕂 𝐴$ is also a Lie algebra. #m/thm/lie

Proof

That the product on $𝔤 \otimes 𝕂 𝐴$ is alternating follows immediately. For the Jacobi identity note

[𝑥 \otimes 𝑎, [𝑦 \otimes 𝑏, 𝑧 \otimes 𝑐]] + [𝑦 \otimes 𝑏, [𝑧 \otimes 𝑐, 𝑥 \otimes 𝑎]] + [𝑧 \otimes 𝑐, [𝑥 \otimes 𝑎, 𝑦 \otimes 𝑏]] = [𝑥, [𝑦, 𝑧]] 𝑎 𝑏 𝑐 + [𝑦, [𝑧, 𝑥]] 𝑏 𝑐 𝑎 + [𝑧, [𝑥, 𝑦]] 𝑐 𝑎 𝑏 = ([𝑥, [𝑦, 𝑧]] + [𝑦, [𝑧, 𝑥]] + [𝑧, [𝑥, 𝑦]]) 𝑎 𝑏 𝑐 = 0

as required.

Note that if $𝐴$ is unital, then we have the injective Lie algebra homomorphism

𝜄 : 𝔤 ↪ 𝔤 \otimes 𝕂 𝐴 𝑥 \mapsto 𝑥 \otimes 1

Functoriality

This construction forms a functor $𝖫 𝗂 𝖾 𝕂 \times 𝖢 𝖠 𝗅 𝗀 𝕂 \to 𝖫 𝗂 𝖾 𝕂$ .

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