On generalized K\"ahler geometry on compact Lie groups

TL;DR

This paper explores the properties of generalized K"ahler structures on compact Lie groups, utilizing spectral sequences to connect bi-Hermitian and generalized geometric data, and clarifies their Hodge and canonical bundle relationships.

Contribution

It introduces a framework for analyzing invariant generalized K"ahler structures on compact Lie groups using spectral sequences and clarifies their geometric and cohomological properties.

Findings

Spectral sequences relate bi-Hermitian data to generalized geometry.

Clarification of generalized Hodge decomposition in this context.

Insights into generalized canonical bundles on Lie groups.

Abstract

We present some fundamental facts about a class of generalized K\"ahler structures defined by invariant complex structures on compact Lie groups. The main computational tool is the BH-to-GK spectral sequences that relate the bi-Hermitian data to generalized geometry data. The relationship between generalized Hodge decomposition and generalized canonical bundles for generalized K\"ahler manifolds is also clarified.