[[Formal sums over a vector space]]
# Formal sums over a Lie algebra

Let $\mathfrak{l}$ be [[Lie algebra]] over $\mathbb{K}$ and consider [[Formal sums over a vector space|formal sums]] $\mathfrak{l}\{ z \}$ over $\mathfrak{l}$.
Then we have the [[Alternating multilinear map|alternating]] [[Multilinear map|bilinear]] map #m/def/fcalc
$$
\begin{align*}
[-,-] : \mathfrak{l}\{z_{1}\} \times \mathfrak{l}\{ z_{2} \} &\to \mathfrak{l}\{ z_{1},z_{2} \}
\end{align*}
$$
so that
$$
\begin{align*}
\left[ \sum_{m \in \mathbb{K}}x_{m}z_{1}^m, \sum_{n \in \mathbb{K}}y_{n}z_{2}^n \right] &= \sum_{m,n \in \mathbb{K}} [x_{m},y_{n}]z_{1}^mz_{2}^n
\end{align*}
$$
where $x_{m},y_{n} \in \mathfrak{l}$ for all $m,n \in \mathbb{K}$.

  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], §2.3, p. 58

## See also

- [[Formal series over an (un)twisted affine Lie algebra]]

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