A Kohno-Drinfeld theorem for the monodromy of cyclotomic KZ connections

TL;DR

This paper explicitly computes the monodromy of cyclotomic KZ connections, linking algebraic quantum group representations with analytic monodromy representations of the type B braid group.

Contribution

It establishes a Kohno-Drinfeld type theorem for cyclotomic KZ connections, connecting algebraic and analytic monodromy representations for type B braid groups.

Findings

Explicit formulas for monodromy representations of cyclotomic KZ systems.

Demonstration of how quantum group representations extend to type B braid groups.

Identification of algebraic and analytic monodromy representations.

Abstract

We compute explicitly the monodromy representations of "cyclotomic" analogs of the Knizhnik--Zamolodchikov differential system. These are representations of the type B braid group $B_n^1$. We show how the representations of the braid group $B_n$ obtained using quantum groups and universal $R$-matrices may be enhanced to representations of $B_n^1$ using dynamical twists. Then, we show how these "algebraic" representations may be identified with the above "analytic" monodromy representations.