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Quantum Drinfeld Hecke Algebras

  Published:2013-06-14
 Printed: Aug 2014
  • Viktor Levandovskyy,
    Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, D-52062 Aachen, Germany
  • Anne V. Shepler,
    Department of Mathematics, University of North Texas, Denton, Texas 76203, USA
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Abstract

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 PBW conditions.
Keywords: skew polynomial rings, noncommutative Gröbner bases, graded Hecke algebras, symplectic reflection algebras, Hochschild cohomology skew polynomial rings, noncommutative Gröbner bases, graded Hecke algebras, symplectic reflection algebras, Hochschild cohomology
MSC Classifications: 16S36, 16S35, 16S80, 16W20, 16Z05, 16E40 show english descriptions Ordinary and skew polynomial rings and semigroup rings [See also 20M25]
Twisted and skew group rings, crossed products
Deformations of rings [See also 13D10, 14D15]
Automorphisms and endomorphisms
Computational aspects of associative rings [See also 68W30]
(Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
16S36 - Ordinary and skew polynomial rings and semigroup rings [See also 20M25]
16S35 - Twisted and skew group rings, crossed products
16S80 - Deformations of rings [See also 13D10, 14D15]
16W20 - Automorphisms and endomorphisms
16Z05 - Computational aspects of associative rings [See also 68W30]
16E40 - (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
 

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