loop algebra

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

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

where $\left[\,,\,\right]_{\mathfrak{g}}$ denotes the Lie bracket from $\mathfrak{g}$.

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 $\mathfrak{g}$ tensored with a power of $t$ and thus is zero in $\mathcal{L}(\mathfrak{g})$.

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 $\mathbb{K}$ 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 $\mathfrak{g}$. We may make this set into a group by defining multiplication  pointwise: given $a,b\colon S^{1}\to G$, we define $(a\cdot b)(x)=a(x)\cdot b(x)$.

References

• 1 Victor Kac, Infinite Dimensional Lie Algebras, Third edition. Cambridge University Press, Cambridge, 1990.
Title loop algebra LoopAlgebra 2013-03-22 15:30:07 2013-03-22 15:30:07 rspuzio (6075) rspuzio (6075) 12 rspuzio (6075) Definition msc 22E60 msc 22E65 msc 22E67 loop algebra