Quantization of cohomology in semi-simple Lie algebras

Abstract: Let ${\scriptstyle\gal}$ be a complex, finite-dimensional Lie algebra, and $\Mo$ a contractable neighborhood of a complex homogeneous space on which ${\scriptstyle\gal}$ acts transitively. The present article investigates quantization of $\rH^1({\scriptstyle\gal};\comega(\Mo)/{\scriptstyle\cnums})$ by the condition that the corresponding realization by first-order operators admits a finite-dimensional invariant subspace of functions. The quantization of cohomology phenomenon surfaced during the recent classification of finite-dimensional Lie algebras of first and zero order operators in two complex variables; this classification revealed that quantization holds for all two-dimensional homogeneous spaces.

The present article presents the first known counter-examples to quantization of cohomology; it is shown that quantization can fail even if ${\scriptstyle\gal}$ 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 ${\scriptstyle\gal}$. It is shown that the set of classes in $\rH^1$ 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 $\rH^1$, 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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