Poincaré–Birkhoff–Witt deformations of smash product algebras from Hopf actions on Koszul algebras

Walton, Chelsea; Witherspoon, Sarah

doi:10.2140/ant.2014.8.1701

Cited by 14 publications

(27 citation statements)

References 29 publications

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“…Theorem 4.6.1, for the notions see Sect. 4.6), generalizing the corresponding results in [7,20,21,24].…”

Section: Theorem a U Is A Pbw Deformation Of B If And Only If θ Is Ansupporting

confidence: 72%

“…We remind the reader that the PBW deformations appeared in [2,18,20,21,24] are more general in terms of generating relations. However, the situation in the theorem above occurs very common, and it applies to many interesting algebras, such as symplectic reflection algebras (cf.…”

Section: Theorem a U Is A Pbw Deformation Of B If And Only If θ Is Anmentioning

confidence: 99%

“…Therefore we obtain the following result. Note that if S is a group algebra (or Hopf algebra) and A is a module algebra over S, then the projective resolution P • of the smash product A#S is equivalent to the resolution obtained in [20] (or [24], also [15]) by choosing suitable ϑ and h. …”

Section: Graded Projective Resolutions Of Koszul Algebras With Pdim Smentioning

confidence: 99%

“…Shepler and Witherspoon proved in [20] (more recently [21]) the PBW theorem for any skew group algebra obtained from an action of a finite group on a Koszul algebra. More generally, Walton and Witherspoon [24] proved the PBW theorem for any smash product algebra obtained from a Hopf algebra acting on a Koszul algebra.…”

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confidence: 99%

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PBW deformations of Koszul algebras over a nonsemisimple ring

Oystaeyen

Zhang

2014

Math. Z.

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Let B be a generalized Koszul algebra over a finite dimensional algebra S. We construct a bimodule Koszul resolution of B when the projective dimension of S B equals two. Using this we prove a Poincaré-Birkhoff-Witt (PBW) type theorem for a deformation of a generalized Koszul algebra. When the projective dimension of S B is greater than two, we construct bimodule Koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and Koszul algebras, and then prove the PBW type theorem. The results obtained can be applied to standard Koszul Artin-Schelter Gorenstein algebras in the sense of Minamoto and Mori (Adv Math 226:4061-4095, 2011).

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“…Theorem 4.6.1, for the notions see Sect. 4.6), generalizing the corresponding results in [7,20,21,24].…”

Section: Theorem a U Is A Pbw Deformation Of B If And Only If θ Is Ansupporting

confidence: 72%

Section: Theorem a U Is A Pbw Deformation Of B If And Only If θ Is Anmentioning

confidence: 99%

Section: Graded Projective Resolutions Of Koszul Algebras With Pdim Smentioning

confidence: 99%

mentioning

confidence: 99%

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PBW deformations of Koszul algebras over a nonsemisimple ring

Oystaeyen

Zhang

2014

Math. Z.

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“…The PBW property depends on the map κ in ways which have been studied extensively in the literature, starting with the work of Drinfeld in [Dr], Braverman and Gaitsgory in [BG] and of Etingof and Ginzburg in [EG]. We refer to the survey [SW3], the article [WW1] and the recent article [Kh2] which contains many references in its introduction. The implications for our special case are studied in [FS].…”

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confidence: 99%

Barbasch-Sahi algebras and Dirac cohomology

Flake

2019

Trans. Amer. Math. Soc.

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Abstract. We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld in [Dr]. In the course of studying the orthogonality condition and in analogy to the orthogonal group we show the existence of a pin cover for cocommutative Hopf algebras over C with an orthogonal module or, more generally, pointed cocommutative Hopf algebras over a field of characteristic 0 with an orthogonal module.Following the suggestion of Dan Barbasch and Siddhartha Sahi, we define a Dirac operator and Dirac cohomology for modules of Hopf-Hecke algebras, generalizing those concepts for connected semisimple Lie groups, graded affine Hecke algebras and symplectic reflection algebras. Using the pin cover, we prove a general theorem for a class of Hopf-Hecke algebras which we call BarbaschSahi algebras, which relates the central character of an irreducible module with non-vanishing Dirac cohomology to the central characters occurring in its Dirac cohomology, generalizing a result called "Vogan's conjecture" for connected semisimple Lie groups which was proved in [HP1], analogous results for graded affine Hecke algebras in [BCT] and for symplectic reflection algebras and Drinfeld Hecke algebras in [Ci].

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