Mathematics – Representation Theory
Scientific paper
1998-05-23
Mathematics
Representation Theory
Length: 16 pages. To appear in the Journal of Lie Theory, Volume 8, #2, 1998
Scientific paper
The space of realizations of a finite-dimensional Lie algebra by first order differential operators is naturally isomorphic to H^1 with coefficients in the module of functions. The condition that a realization admits a finite-dimensional invariant subspace of functions seems to act as a kind of quantization condition on this H^1. It was known that this quantization of cohomology holds for all realizations on 2-dimensional homogeneous spaces, but the extent to which quantization of cohomology is true in general was an open question. The present article presents the first known counter-examples to quantization of cohomology; it is shown that quantization can fail even if the Lie algebra is semi-simple, and even if the homogeneous space in question is compact. A explanation for the quantization phenomenon is given in the case of semi-simple Lie algebras. It is shown that the set of classes in H^1 that admit finite-dimensional invariant subspaces is a semigroup that lies inside a finitely-generated abelian group. In order for this abelian group be a discrete subset of H^1, i.e. in order for quantization to take place, some extra conditions on the isotropy subalgebra are required. Two different instances of such necessary conditions are presented.
Milson Robert
Richter Dieter
No associations
LandOfFree
Quantization of cohomology in semi-simple Lie algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Quantization of cohomology in semi-simple Lie algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quantization of cohomology in semi-simple Lie algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-561174