Hopf-theoretic approach to motives of twisted flag varieties

Abstract

Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the Levine–Morel sense. 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.

Comodule Structures, Equivariant Hopf Structures, and Generalized Schubert Polynomials

“…The determination based on computer-assisted computation can be found in Duan and Zhao [5], [6]. The method using motives can be found in Petrov and Semenov [19].…”

$J$-invariant of linear algebraic groups of outer type

“…For example, it plays an important role in the progress on the Kaplansky problem about possible values of the u-invariant of fields by Vishik [Vi07] and in the solution of a problem of Serre about groups of type E 8 and its finite subgroups [S16], [GS10]. More recently, Petrov and Semenov generalized the J-invariant for groups of inner type to arbitrary oriented cohomology theories in the sense of Levine-Morel [LM] satisfying some axioms (see [PS21]).…”

The J-invariant over splitting fields of Tits algebras

“…Another example is the proof of a conjecture of Serre about groups of type E 8 and its finite subgroups, where the J-invariant plays a crucial role (see [GS10] and [Se16]). More recently, Petrov and Semenov generalized the J-invariant for groups of inner type to arbitrary oriented cohomology theories in the sense of Levine-Morel [LM] satisfying some axioms (see [PS21]).…”