dorsal/arxiv
View SchemaQuantization of cohomology in semi-simple Lie algebras
| Authors | R. Milson, D. Richter |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9805013 |
| URL | https://arxiv.org/abs/solv-int/9805013 |
Abstract
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.
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"abstract": "The space of realizations of a finite-dimensional Lie algebra by first order\ndifferential operators is naturally isomorphic to H^1 with coefficients in the\nmodule of functions. The condition that a realization admits a\nfinite-dimensional invariant subspace of functions seems to act as a kind of\nquantization condition on this H^1. It was known that this quantization of\ncohomology holds for all realizations on 2-dimensional homogeneous spaces, but\nthe extent to which quantization of cohomology is true in general was an open\nquestion. The present article presents the first known counter-examples to\nquantization of cohomology; it is shown that quantization can fail even if the\nLie algebra is semi-simple, and even if the homogeneous space in question is\ncompact. A explanation for the quantization phenomenon is given in the case of\nsemi-simple Lie algebras. It is shown that the set of classes in H^1 that admit\nfinite-dimensional invariant subspaces is a semigroup that lies inside a\nfinitely-generated abelian group. In order for this abelian group be a discrete\nsubset of H^1, i.e. in order for quantization to take place, some extra\nconditions on the isotropy subalgebra are required. Two different instances of\nsuch necessary conditions are presented.",
"arxiv_id": "solv-int/9805013",
"authors": [
"R. Milson",
"D. Richter"
],
"categories": [
"solv-int",
"math.RT",
"nlin.SI"
],
"title": "Quantization of cohomology in semi-simple Lie algebras",
"url": "https://arxiv.org/abs/solv-int/9805013"
},
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