loop algebra

Let $𝔤$ be a Lie algebra over a field $𝕂$. The loop algebra based on $𝔤$ is defined to be $ℒ(𝔤):=𝔤{\otimes }_{𝕂}𝕂[t,t^{-1}]$ as a vector space over $𝕂$. The Lie bracket is determined by

$$[X\otimes t^k,Y\otimes t^l]={[X,Y]}_{𝔤}\otimes t^{k+l}$$

where ${[,]}_{𝔤}$ denotes the Lie bracket from $𝔤$.

This clearly determines a Lie bracket. For instance the three term sum in the Jacobi identity (for elements which are homogeneous in $t$) simplifies to the three term sum for the Jacobi identity in $𝔤$ tensored with a power of $t$ and thus is zero in $ℒ(𝔤)$.

The name “loop algebra” comes from the fact that this Lie algebra arises in the study of Lie algebras of loop groups. For the time being, assume that $𝕂$ is the real or complex numbers so that the familiar structures of analysis and topology are available. Consider the set of all mappings from the circle $S^1$ (we may think of this circle more concretely as the unit circle of the complex plane) to a finite-dimensional Lie group $G$ with Lie algebra is $𝔤$. We may make this set into a group by defining multiplication pointwise: given $a,b:S^1\to G$, we define $(a\cdot b)(x)=a(x)\cdot b(x)$.