Un scindage du morphisme de Frobenius quantique

TL;DR

This paper demonstrates a multiplicative splitting of Lusztig's quantum Frobenius morphism at roots of unity, and constructs orthogonal idempotent bases in related quantum and algebraic structures.

Contribution

It introduces a non-unital splitting of the quantum Frobenius morphism and constructs orthogonal idempotent bases in small quantum and algebraic distribution algebras.

Findings

Existence of a multiplicative splitting of the quantum Frobenius morphism.

Construction of orthogonal idempotent bases in the toral part of the small quantum algebra.

Orthogonal idempotent bases in the algebra of distributions for semisimple algebraic groups.

Abstract

We show that the quantum Frobenius morphism constructed by Lusztig in the setting of the quantum enveloping algebra specialized at a root of unity admits a multiplicative splitting (non unital). We also find a basis of the toral part of the small quantum algebra consisting of pairwise orthogonal idempotents summing up to 1, and likewise in the modular case of the algebra of distributions for a semisimple algebraic group.