The J-invariant, Tits algebras and Triality

TL;DR

This paper explores the relationship between Tits algebra indices and the motivic J-invariant of algebraic groups, providing new insights and explicit classifications for algebras with orthogonal involution up to degree 8.

Contribution

It establishes a connection between Tits algebra indices and the J-invariant, and offers explicit classifications and examples for algebras with orthogonal involution.

Findings

Classified all possible J-invariant values up to degree 8.

Connected Tits algebra indices with the J-invariant.

Provided explicit examples of algebras with orthogonal involution.

Abstract

In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group $G$ and the degree one parameters of its motivic $J$-invariant. Our main technical tool are the second Chern class map and Grothendieck's $\gamma$-filtration. As an application we recover some known results on the $J$-invariant of quadratic forms of small dimension; we describe all possible values of the $J$-invariant of an algebra with orthogonal involution up to degree 8 and give explicit examples; we establish several relations between the $J$-invariant of an algebra $A$ with orthogonal involution and the $J$-invariant of the corresponding quadratic form over the function field of the Severi-Brauer variety of $A$.