where denotes the Lie bracket from .
This clearly determines a Lie bracket. For instance the three term sum in the Jacobi identity (for elements which are homogeneous in ) simplifies to the three term sum for the Jacobi identity in tensored with a power of and thus is zero in .
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 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 (we may think of this circle more concretely as the unit circle of the complex plane) to a finite-dimensional Lie group with Lie algebra is . We may make this set into a group by defining multiplication pointwise: given , we define .
- 1 Victor Kac, Infinite Dimensional Lie Algebras, Third edition. Cambridge University Press, Cambridge, 1990.
|Date of creation||2013-03-22 15:30:07|
|Last modified on||2013-03-22 15:30:07|
|Last modified by||rspuzio (6075)|