Loop algebra

Remarks

On the basis elements $x \otimes t^m$ the bracket reads $[x \otimes t^m, y \otimes t^n] = [x, y] \otimes t^{m+n}$. It is antisymmetric and satisfies the Jacobi identity because the bracket of $\mathfrak{g}$ does and $k[t, t^{-1}]$ is commutative and associative.

Example: for $\mathfrak{g} = \mathfrak{sl}_2$ with basis $e, f, h$, the loop algebra has basis $e \otimes t^m$, $f \otimes t^m$, $h \otimes t^m$ for $m \in \Z$. For instance $[e \otimes t, f \otimes t^{-1}] = h \otimes 1$.

Non-example: in the affine Lie algebra $\widehat{\mathfrak{sl}}_2$ one has $[e \otimes t, f \otimes t^{-1}] = h \otimes 1 + (e \mid f)\, c$ with a nonzero central element $c$ and a nonzero invariant form value $(e \mid f)$. This bracket is not of the form $[x, y] \otimes fg$, so the affine algebra is a central extension of the loop algebra and not a loop algebra itself.

The subalgebra $\mathfrak{g} \otimes k[t]$ is often called the current algebra $\mathfrak{g}[t]$. Over $\C$ one often writes $L\mathfrak{g}$ for this algebra of polynomial loops and uses other names for completions, such as smooth loops $S^1 \to \mathfrak{g}$.