Hopf-theoretic approach to motives of twisted flag varieties

TL;DR

This paper introduces a Hopf algebra-based method to analyze the motives of twisted flag varieties within oriented cohomology theories, offering a unified framework for understanding their structure.

Contribution

It develops a uniform approach to motives of geometrically cellular smooth projective G-varieties using Hopf algebra structures, with applications to twisted flag varieties.

Findings

Established a Hopf algebra framework for motives

Applied the method to twisted flag varieties

Provided new structural insights into motives

Abstract

Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the sense of Levine--Morel. We provide a uniform approach to the $A^*$-motives of geometrically cellular smooth projective $G$-varieties based on the Hopf algebra structure of $A^*(G)$. Using this approach we provide various applications to the structure of motives of twisted flag varieties.