[[Lie algebras MOC]]
# Loop algebra

Let $\mathfrak{g}$ be a [[Lie algebra]] over $\mathbb{K}$. 
The **loop algebra** $\mathfrak{g}[t,t^{-1}]$ of $\mathfrak{g}$ is the [[Tensor product of a Lie algebra and a commutative algebra|tensor product algebra]] $\mathfrak{g} \otimes \mathbb{K}[t,t^{-1}]$ where $\mathbb{K}[t,t^{-1}]$ is the [[algebra of Laurent polynomials]], #m/def/lie 
i.e. with the bracket
$$
\begin{align*}
[x \otimes f, y \otimes g] = [x,y] \otimes fg
\end{align*}
$$
for any $x,y \in \mathfrak{g}$ and $f,g \in \mathbb{K}[t,t^{-1}]$.
This may also be viewed as [[Formal sums over a Lie algebra|formal series]] $\mathfrak{g}[t,t^{-1}]$.

## See also

- [[Affine Lie algebra]]
- [[Twisted affine Lie algebra]]

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