Koszulness of Enveloping Algebras Associated to Generalized Yang-Baxter Equations

TL;DR

This paper proves that certain universal enveloping algebras linked to generalized Yang-Baxter equations are Koszul, extending known results to new algebra series and exploring their implications for Cherednik algebra representations.

Contribution

It establishes the Koszulness of enveloping algebras for the Bn and Dn series, expanding the class of known Koszul algebras related to generalized Yang-Baxter equations.

Findings

Enveloping algebras for Bn and Dn series are Koszul.

Results enable construction of adjoint functors between Cherednik algebra categories.

Extends Koszul property to broader classes of braided Hopf algebras.

Abstract

The universal enveloping algebra $U(\mathfrak{tr}_n)$ of a Lie algebra associated to the classical Yang-Baxter equation was introduced in [BEER06] where it was shown to be Koszul. This algebra appears as the $A_{n-1}$ case in a general class of braided Hopf algebras in [BB09] for any complex reflection group. In this paper, we show that the algebras corresponding to the series $B_n$ and $D_n$, which are again universal enveloping algebras, are Koszul. We further show how results of [BB09] can be used to produce pairs of adjoint functors between categories of rational Cherednik algebra representations of different rank and type for the classical series of Coxeter groups.