[[Algebra theory MOC]]
# Tensor product of algebras

Let $(A, \cdot_{A})$ and $(B, \cdot_{B})$ be $\mathbb{K}$-[[K-algebra|algebras]]
The **tensor product algebra** $(A \otimes B, \cdot_{A \otimes B})$ is their [[Tensor product of vector spaces|tensor product vector space]] $A \otimes B$ along with the product defined by the following commutative diagram #m/def/falg 

![[Tensor product of algebras.svg#invert|https://q.uiver.app/#q=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]]

where $\beta : a \otimes b \mapsto b \otimes a$ is the braiding morphism for $\Vect_{\mathbb{K}}$.
Thus
$$
\begin{align*}
(a_{1} \otimes b_{1}) \cdot (a_{2} \otimes b_{2}) = (a_{1}\cdot a_{2}) \otimes (b_{1} \cdot b_{2})
\end{align*}
$$

## Special cases

- [[Tensor product of a Lie algebra and a commutative algebra]] is a Lie algebra

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