Duality properties for quantum groups

TL;DR

This paper explores duality properties in quantum groups and deformation Hopf algebras, extending classical dualities to quantum settings and establishing Poincaré duality under certain conditions.

Contribution

It generalizes duality properties from enveloping algebras to deformation Hopf algebras, including quantum groups, and proves Poincaré duality in this context.

Findings

Duality properties extend to deformation Hopf algebras

The character of $A_h$ lifts the classical $Trad_{\goth g}$ character

Poincaré duality holds for deformation Hopf algebras with finite homological dimension

Abstract

Some duality properties for induced representations of enveloping algebras involve the character $Trad_{\goth g}$. We extend them to deformation Hopf algebras $A_{h}$ of a noetherian Hopf $k$-algebra $A_{0}$ satistying $Ext^{i}_{A_{0}}(k, A_{0})=\{0\}$ except for $i=d$ where it is isomorphic to $k$. These duality properties involve the character of $A_{h}$ defined by right multiplication on the one dimensional free $k[[h]]$-module $Ext^{d}_{A_{h}} (k[[h]], A_{h})$. In the case of quantized enveloping algebras, this character lifts the character $Trad_{\goth g}$. We also prove Poincar{\'e} duality for such deformation Hopf algebras in the case where $A_{0}$ is of finite homological dimension. We explain the relation of our construction with quantum duality.