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Quantum Drinfeld Hecke Algebras

Published online by Cambridge University Press:  20 November 2018

Viktor Levandovskyy  and
Anne V. Shepler
Viktor Levandovskyy
Affiliation:
Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, D-52062 Aachen, Germany. e-mail: levandov@math.rwth-aachen.de
Anne V. Shepler
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203, USA. e-mail: ashepler@unt.edu
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Abstract

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We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré–Birkhoff–Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincaré–Birkhoff–Witt conditions.

Keywords

16S3616S3516S8016W2016Z0516E40skew polynomial ringsnoncommutative Gröbner basesgraded Hecke algebrassymplectic reflection algebrasHochschild cohomology
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 4 , 01 August 2014 , pp. 874 - 901
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

*

Work of the second author was partially supported by National Science Foundation research grants #DMS-0800951 and #DMS-1101177 and a research fellowship from the Alexander von Humboldt Foundation.

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