Duality properties for quantum groups

Details Duality properties for quantum groups Duality properties for quantum groups

2009-11-15

arxiv.org/abs/0911.2860v1

Mathematics

Quantum Algebra

38 pages

Scientific paper

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.

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