The Looijenga-Lunts-Verbitsky algebra and Verbitsky's Theorem
Alessio Bottini
TL;DR
This paper reviews the structure of the LLV Lie algebra and its action on the rational cohomology of compact Kähler and hyperkähler manifolds, highlighting its algebraic properties and specific cohomology components.
Contribution
It provides a detailed analysis of the LLV Lie algebra's structure and describes a key irreducible component of the cohomology for hyperkähler manifolds.
Findings
Characterization of the LLV Lie algebra structure
Description of a specific cohomology component in hyperkähler manifolds
Insights into the algebra's action on rational cohomology
Abstract
In these notes we review some basic facts about the LLV Lie algebra. It is a rational Lie algebra, introduced by Looijenga-Lunts and Verbitsky, acting on the rational cohomology of a compact K\"{a}hler manifold. We study its structure and describe one irreducible component of the rational cohomology in the case of a compact hyperk\"{a}hler manifold.
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models