The crystal commutor and Drinfeld's unitarized R-matrix

TL;DR

This paper compares Drinfeld's unitarized R-matrix with Henriques and Kamnitzer's crystal commutor, showing their equivalence in a specific case and analyzing their actions on crystal bases.

Contribution

It establishes the equivalence between Drinfeld's and Henriques-Kamnitzer's commutors in a particular case and explores their effects on crystal bases.

Findings

Drinfeld's unitarized R-matrix matches Henriques-Kamnitzer's commutor in a specific case.

The paper describes how Drinfeld's commutor acts on tensor products of crystal bases.

It clarifies the relationship between the crystal commutor and Drinfeld's construction.

Abstract

Drinfeld defined a unitarized R-matrix for any quantum group U_q(g). This gives a commutor for the category of U_q(g) representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives U_q(g) representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer's construction agrees with Drinfeld's commutor. We then describe the action of Drinfeld's commutor on a tensor product of two crystal bases, and explain the relation to the crystal commutor.