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
Canonical name	LoopAlgebra
Date of creation	2013-03-22 15:30:07
Last modified on	2013-03-22 15:30:07
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	12
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 22E60
Classification	msc 22E65
Classification	msc 22E67
Defines	loop algebra