# The Borel character

## TL;DR

This paper introduces the Borel character, a quadratic analog of the Chern character, linking rational higher Grothendieck-Witt groups with MW-motivic and motivic cohomologies, and explores related ternary laws.

## Contribution

It defines the Borel character, computes additive ternary laws, and demonstrates an embedding of Milnor-Witt K-theory into higher Grothendieck-Witt groups.

## Key findings

- Borel character links Grothendieck-Witt groups with cohomologies.
- Computed additive ternary laws for MW-motivic cohomology.
- Embedded Milnor-Witt K-theory into higher Grothendieck-Witt groups.

## Abstract

The main purpose of this article is to define a quadratic analog of the Chern character, the so-called Borel character, which identifies rational higher Grothendieck-Witt groups with a sum of rational MW-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analog of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendieck-Witt groups of F modulo explicit torsion.