Twisted Hochschild homology of quantum flag manifolds: 2-cycles from invariant projections

We study the twisted Hochschild homology of quantum full flag manifolds, with the twist being the modular automorphism of the Haar state. We show that nontrivial [Formula: see text]-cycles can be constructed from appropriate invariant projections. Moreover, we show that [Formula: see text] has dimension at least [Formula: see text]. We also discuss the case of generalized flag manifolds and present the example of the quantum Grassmannians.

PARABOLIC SUBROOT SYSTEMS AND THEIR APPLICATIONS

In this note we consider parabolic subroot systems of a complex simple Lie Algebra. We describe root theoretic data of the subroot systems in terms of that of the root system and we give a selection of applications of our results to the study of generalized flag manifolds.

Invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$

It is well known that the Einstein equation on a Riemannian flag manifold $(G/K,g)$ reduces to an algebraic system if $g$ is a $G$-invariant metric. In this paper we obtain explicitly new invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$; and we compute the Einstein system for generalized flag manifolds of type $Sp(n)$. We also consider the isometric problem for these Einstein metrics.

Toric Degenerations, Tropical Curve, and Gromov–Witten Invariants of Fano Manifolds

In this paper, we give a tropical method for computing Gromov–Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds that admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A and some moduli space of rank two bundles on a genus two curve.