Hausdorffness of certain nilpotent cohomology spaces
Fabian Januszewski, Binyong Sun, Hao Ying

TL;DR
This paper proves that certain nilpotent Lie algebra cohomology and homology spaces associated with smooth representations of compact Lie groups are Hausdorff, ensuring well-behaved topological properties in these algebraic structures.
Contribution
It establishes the Hausdorff property for Lie algebra cohomology and homology spaces in the context of smooth group representations, a result not previously confirmed.
Findings
Cohomology spaces are Hausdorff.
Homology spaces are Hausdorff.
Results apply to nilpotent radicals of parabolic subalgebras.
Abstract
Let be a smooth representation of a compact Lie group on a quasi-complete locally convex complex topological vector space. We show that the Lie algebra cohomology space and the Lie algebra homology space are both Hausdorff, where is the nilpotent radical of a parabolic subalgebra of the complexified Lie algebra of .
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Advanced Topology and Set Theory
