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Degree-One Rational Cherednik Algebras for the Symmetric Group
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $H_{0,c}$.
Keywords: rational Cherednik algebra, skew group algebra, Drinfeld orbifold algebra, Hochschild cohomology, symmetric group.
Mots-clés : deformations, Poincaré–Birkhoff–Witt conditions 
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     author = {Briana Foster-Greenwood and Cathy Kriloff},
     title = {Degree-One {Rational} {Cherednik} {Algebras} for the {Symmetric} {Group}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2021},
     volume = {17},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a38/}
}
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Briana Foster-Greenwood; Cathy Kriloff. Degree-One Rational Cherednik Algebras for the Symmetric Group. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a38/

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