Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 039, 35 pages      arXiv:1912.06743      https://doi.org/10.3842/SIGMA.2021.039

Degree-One Rational Cherednik Algebras for the Symmetric Group

Briana Foster-Greenwood a and Cathy Kriloff b
a) Department of Mathematics and Statistics, California State Polytechnic University, Pomona, California 91768, USA
b) Department of Mathematics and Statistics, Idaho State University, Pocatello, Idaho 83209, USA

Received August 07, 2020, in final form April 02, 2021; Published online April 19, 2021

Abstract
Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of $\mathfrak{gl}_n$-type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the $\mathfrak{sl}_n$-type rational Cherednik algebras $H_{0,c}$.

Key words: rational Cherednik algebra; skew group algebra; deformations; Drinfeld orbifold algebra; Hochschild cohomology; Poincaré-Birkhoff-Witt conditions; symmetric group.

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