Tensor product of algebras

Let $(𝐴, \cdot 𝐴)$ and $(𝐵, \cdot 𝐵)$ be $𝕂$ -algebras The tensor product algebra $(𝐴 \otimes 𝐵, \cdot 𝐴 \otimes 𝐵)$ is their tensor product vector space $𝐴 \otimes 𝐵$ along with the product defined by the following commutative diagram #m/def/ralg

where $𝛽 : 𝑎 \otimes 𝑏 \mapsto 𝑏 \otimes 𝑎$ is the braiding morphism for $𝖵 𝖾 𝖼 𝗍 𝕂$ . Thus

(𝑎 1 \otimes 𝑏 1) \cdot (𝑎 2 \otimes 𝑏 2) = (𝑎 1 \cdot 𝑎 2) \otimes (𝑏 1 \cdot 𝑏 2)

Special cases

Tensor product of a Lie algebra and a commutative algebra is a Lie algebra

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