Koszul duality for semidirect products and generalized Takiff algebras

TL;DR

This paper establishes Koszul dualities for categories of graded modules over certain semidirect product algebras, including universal enveloping algebras of Takiff Lie algebras and quantum group deformations.

Contribution

It introduces a general framework for Koszul duality applicable to semidirect product algebras arising from bialgebras and graded module algebras, extending to quantum groups.

Findings

Koszul dualities for graded modules over semidirect product algebras.

Application to universal enveloping algebras of Takiff Lie algebras.

Extension to quantum groups and their deformations.

Abstract

We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the latter. In particular, this applies to graded representations of the universal enveloping algebra of the Takiff Lie algebra (or the truncated current algebra) and its (super)analogues, and also to semidirect products of quantum groups with braided symmetric and exterior module algebras in case the latter are flat deformations of classical ones.