Motivic equivalence of algebraic groups

TL;DR

This paper establishes criteria for motivic equivalence of semisimple algebraic groups using higher Tits p-indices, enabling classification of classical groups and linking motivic classes to stable birational equivalence of Severi-Brauer varieties.

Contribution

It provides a comprehensive classification of absolutely simple classical groups up to motivic equivalence based on algebraic structures and introduces criteria using higher Tits p-indices.

Findings

Complete classification of classical groups up to motivic equivalence.

Criteria for motivic equivalence based on higher Tits p-indices.

Bijection between stable birational classes of Severi-Brauer varieties and motivic classes of projective linear groups.

Abstract

Two semisimple algebraic groups of the same type are said to be motivic equivalent if the motives of the associated projective homogeneous varieties of the same type are isomorphic. We give general criteria of motivic equivalence in terms of the so-called higher Tits p-indices of algebraic groups. These results allow to give a complete classification of absolutely simple classical groups up to motivic equivalence in terms of the underlying algebraic structures. Among other applications of this classification, we deduce that there is a bijection between the stable birational equivalence classes of Severi-Brauer varieties of fixed dimension and the motivic equivalence classes of projective linear groups of the same rank.