On motives of algebraic groups associated to a divsion algebra

TL;DR

This paper explores the motives of algebraic groups linked to division algebras, connecting their motives to those of associated Severi-Brauer varieties within the framework of motivic complexes.

Contribution

It establishes a relationship between motives of algebraic groups from division algebras and motives of specific simplicial schemes, extending Suslin's ideas.

Findings

Motives of GL_1(A) and SL_1(A) are related to the motive of the Severi-Brauer variety.

The work connects algebraic group motives to Voevodsky's simplicial schemes.

Provides a new perspective on motives associated with division algebras.

Abstract

We consider algebraic groups GL_1(A), SL_1(A), where A is a division algebra of prime degree over a field F, and associated motives in the category of motivic complexes DM(F). Following an idea of Suslin, we relate motives of these groups to the motive of Voevodsky's simplicial scheme X, associated to the Severi-Brauer variety of A.