## ALGEBRAS AND THEIR PBW DEFORMATIONS

### Citations

277 | Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, arXiv:math/0011114
- Etingof, Ginzburg
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Citation Context ...and Gaitsgory gave necessary and sufficient conditions for a non-homogeneous quadratic algebra to be a Poincaré-Birkoff-Witt (PBW for short) deformation of a Koszul algebra [1]. Etingof and Ginzburg =-=[6]-=- extended the conditions formulated by Braverman and Gaitsgory to Koszul algebras over semisimple algebras (for example, over a group algebra CG). They used this generalization to investigate the symp... |

122 |
Oystaeyen, Graded ring theory
- Năstăsescu, Van
- 1982
(Show Context)
Citation Context ... 0 such that each Pi is finitely generated as B e-module. The algebra U e is a bounded below filtered algebra and U is a bounded below filtered U e-module. By Theorem D.VII.5 and Corollary D.VII.6 in =-=[17]-=-, U admits a filtered projective U e-module resolution: 0→ P ′d → P ′ d−1 → · · · → P ′ 1 → P ′ 0 → U → 0 such that its associated graded resolution is the resolution (28). Hence each P ′i is finitely... |

85 | The Poincaré-Brkhoff-Witt theorem for quadratic algebras of Koszul type, J.Algebra 181
- Braverman, Gaistgory
- 1996
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Citation Context ...uction In 1996, Braverman and Gaitsgory gave necessary and sufficient conditions for a non-homogeneous quadratic algebra to be a Poincaré-Birkoff-Witt (PBW for short) deformation of a Koszul algebra =-=[1]-=-. Etingof and Ginzburg [6] extended the conditions formulated by Braverman and Gaitsgory to Koszul algebras over semisimple algebras (for example, over a group algebra CG). They used this generalizati... |

76 |
A Relation between Hochschild Homology and Cohomology for Gorenstein
- Bergh
- 1998
(Show Context)
Citation Context ... sequence converges. In conclusion, we obtain that ExtiUe(U,U e) = 0 for i 6= d1 + d2 and Ext d1+d2 Ue (U,U e) = W , where W is a filtered U e-module such that Gr(W ) ∼= Bρ. By Lemma 4 and Lemma 5 in =-=[21]-=-, we obtain that W = Uψ, where ψ is a filtration-preserving automorphism of U such that Gr(ψ) = ρ. Finally, we show that U is homologically smooth. Note that the algebra B is a twisted graded CY algeb... |

43 | N.Marconnet, Koszul and Gorenstein properties for homogeneous algebras - Berger |

31 |
duality, localization, and smash products
- Farinati, Hochschild
- 2005
(Show Context)
Citation Context ...a, where k〈µ〉 is the free abelian group generated by µ. A more general result has been proved in [9] using a different method. Other results on the CY property of smash products can be found, e.g. in =-=[7]-=-, [24] and [26]. In Section 3, we study the CY property of PBW deformations. The motivating examples come from pointed Hopf algebras of finite Cartan type. In [27], the authors showed that the lifting... |

27 | Poincare-BirkhoffWitt deformations of Calabi-Yau algebras
- Berger, Taillefer
- 2007
(Show Context)
Citation Context ...ash product algebra obtained from a braiding between a finite dimensional algebra and a Koszul algebra, generalizing the corresponding result in [20] and [23]. For a Koszul algebra, it was studied in =-=[3]-=- and [25] when the Calabi-Yau (CY for short) property is preserved under PBW deformation. For certain CY Hopf algebras, it was found that their PBW deformations are still CY algebras 2010 Mathematics ... |

22 | Some finite-dimensional algebras related to elliptic curves. In Representation theory of algebras and related topics (Mexico City - Smith - 1994 |

22 |
Noncommutative homology of some three-dimensional quantum spaces
- Bergh
- 1992
(Show Context)
Citation Context ...gree i for all i > 0. Now let A = T (V )/〈R〉 be a Koszul algebra. Let {ei}i=1,2,··· ,n be a basis of V and {e∗i }i=1,2,··· ,n the dual basis. We recall here the Koszul bimodule complex constructed in =-=[22]-=-: Kb(A) : · · · → A⊗ (A ! p) ∗⊗A dp −→ A⊗ (A!p−1) ∗⊗A→ · · · → A⊗V ⊗A d1−→ A⊗A→ 0, where the differential dp is defined by dp(x⊗ ω ⊗ y) = n∑ i=1 xei ⊗ e ∗ i · ω ⊗ y + (−1) q n∑ i=1 x⊗ ω · e∗i ⊗ eiy, f... |

16 |
Gorenstein subrings of invariants under Hopf algebra actions
- Kirkman, Kuzmanovich, et al.
(Show Context)
Citation Context ...opf algebra, A a p-Koszul graded CY algebra such that A is an H-module algebra. In [16], the authors proved that A#H is CY if and only if the homological determinant (for the definition, see for e.g. =-=[14]-=-) of the Hopf action on A is trivial. In this section, we want to show that the result still holds when H is not involutory. The key point is to introduce the algebra “Ae⋊SiH” (see definition 2.1 for ... |

13 | and representations of infinitesimal Hecke algebras of sl2
- Tikaradze, Center
- 2008
(Show Context)
Citation Context ...isimple or even infinite dimensional. Particular examples of such algebras are smash products of Koszul algebras with Hopf algebras. In literature, some specific cases have been studied, e.g. in [8], =-=[15]-=-, [20] etc.. Recently, Walton and Witherspoon [23] proved the PBW theorem for any smash product algebra obtained from a Hopf algebra acting on a Koszul algebra. In [11], He, Van Oystaeyen and Zhang pr... |

10 | Quantized symplectic oscillator algebras of rank one
- Gan, Khare
(Show Context)
Citation Context ...n-semisimple or even infinite dimensional. Particular examples of such algebras are smash products of Koszul algebras with Hopf algebras. In literature, some specific cases have been studied, e.g. in =-=[8]-=-, [15], [20] etc.. Recently, Walton and Witherspoon [23] proved the PBW theorem for any smash product algebra obtained from a Hopf algebra acting on a Koszul algebra. In [11], He, Van Oystaeyen and Zh... |

9 | Cocommutative Calabi-Yau Hopf Algebras and Deformations”,
- He, Oystaeyen, et al.
- 2010
(Show Context)
Citation Context ...ct Classification. 16E65, 16S37, 16S40, 16S80 . Key words and phrases. Koszul algebras, Calabi-Yau algebras, smash products, PBW deformation . 1 2 XIAOLAN YU, FRED VAN OYSTAEYEN, AND YINHUO ZHANG cf. =-=[10]-=-, [12] [27]. The aim of this paper is to investigate the Calabi-Yau property of the PBW deformation of the algebra A#H, where A is a Koszul CY algebra and H is a CY Hopf algebra acting on A. To this a... |

9 | Skew Calabi-Yau Algebras and Homological Identities”, arXiv:1302.0437v2 [math.RA], - Reyes, Rogalski, et al. - 2013 |

8 | Rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators - Chemla |

8 | A Poincaré-Birkoff-Witt theorem for quadratic algebras with group actions
- Shepler, Witherspoon
(Show Context)
Citation Context ...e or even infinite dimensional. Particular examples of such algebras are smash products of Koszul algebras with Hopf algebras. In literature, some specific cases have been studied, e.g. in [8], [15], =-=[20]-=- etc.. Recently, Walton and Witherspoon [23] proved the PBW theorem for any smash product algebra obtained from a Hopf algebra acting on a Koszul algebra. In [11], He, Van Oystaeyen and Zhang proved t... |

8 |
Skew group algebras of Calabi-Yau algebras”,
- Wu, Zhu
- 2011
(Show Context)
Citation Context ...ere k〈µ〉 is the free abelian group generated by µ. A more general result has been proved in [9] using a different method. Other results on the CY property of smash products can be found, e.g. in [7], =-=[24]-=- and [26]. In Section 3, we study the CY property of PBW deformations. The motivating examples come from pointed Hopf algebras of finite Cartan type. In [27], the authors showed that the liftings of a... |

8 |
Poincaré-Birkhoff-Witt deformation of Koszul Calabi-Yau algebras, Algebra and Representation Theory
- Wu, Zhu
- 2011
(Show Context)
Citation Context ...uct algebra obtained from a braiding between a finite dimensional algebra and a Koszul algebra, generalizing the corresponding result in [20] and [23]. For a Koszul algebra, it was studied in [3] and =-=[25]-=- when the Calabi-Yau (CY for short) property is preserved under PBW deformation. For certain CY Hopf algebras, it was found that their PBW deformations are still CY algebras 2010 Mathematics Subject C... |

6 |
Dualizing complexes and twisted Hochschild (co)homology for noetherian Hopf algebras
- Brown, Zhang
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Citation Context ..., (iv) A is of finite global dimension d, then A is called AS-regular. Lemma 1.4. [18, Lemma 1.2] Let A be a connected graded algebra . Then A is graded twisted CY if and only if it is AS-regular. In =-=[4]-=-, an AS-Gorenstein (regular) algebra can be defined on an augmented algebra in general. Definition 1.5. (cf. [4, defn. 1.2]). (i) Let A be a left Noetherian augmented algebra with a fixed augmentation... |

6 | Hopf-Hochschild (co)homology of module algebras
- Kaygun
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Citation Context ...⊗ h) = a(Sig1) · a ′ ⊗ (g3 · b ′)b⊗ g2h, for any a⊗ b⊗ g, a′ ⊗ b′ ⊗ h ∈ A⊗A⊗H. KOSZUL TWISTED CALABI-YAU ALGEBRAS 7 Remark 2.2. (i) When i = 0, Ae⋊Si H is just the algebra A e ⋊H introduced by Kaygun =-=[13]-=-. (ii) An Ae ⋊Si H-module M is a vector space such that it is both an A emodule and an H-module satisfying, for any h ∈ H, a, b ∈ A and m ∈M , (2) h · (amb) = ((Sih1) · a)(h2 ·m)(h3 · b). Lemma 2.3. L... |

6 | Calabi-Yau pointed Hopf algebras of finite Cartan type”,
- Yu, Zhang
- 2013
(Show Context)
Citation Context ...cation. 16E65, 16S37, 16S40, 16S80 . Key words and phrases. Koszul algebras, Calabi-Yau algebras, smash products, PBW deformation . 1 2 XIAOLAN YU, FRED VAN OYSTAEYEN, AND YINHUO ZHANG cf. [10], [12] =-=[27]-=-. The aim of this paper is to investigate the Calabi-Yau property of the PBW deformation of the algebra A#H, where A is a Koszul CY algebra and H is a CY Hopf algebra acting on A. To this aim, we firs... |

4 | Poincaré-Birkhoff-Witt deformations of smash product algebras from Hopf actions on Koszul algebras, arXiv:1308.6011
- Walton, Witherspoon
(Show Context)
Citation Context ...xamples of such algebras are smash products of Koszul algebras with Hopf algebras. In literature, some specific cases have been studied, e.g. in [8], [15], [20] etc.. Recently, Walton and Witherspoon =-=[23]-=- proved the PBW theorem for any smash product algebra obtained from a Hopf algebra acting on a Koszul algebra. In [11], He, Van Oystaeyen and Zhang proved the PBW theorem for a generalized smash produ... |

3 | Untwisting a twisted Calabi-Yau algebra
- Goodman, Kraehmer
(Show Context)
Citation Context ...or a Koszul twisted CY algebra A with Nakayama automorphism µ, the algebra A#k〈µ〉 is always a CY algebra, where k〈µ〉 is the free abelian group generated by µ. A more general result has been proved in =-=[9]-=- using a different method. Other results on the CY property of smash products can be found, e.g. in [7], [24] and [26]. In Section 3, we study the CY property of PBW deformations. The motivating examp... |

2 |
Hopf Action on Calabi-Yau algebras, New trends in noncommutative algebra
- Liu, Wu, et al.
- 2012
(Show Context)
Citation Context ... and only if ξ = ε, and S2 is an inner automorphism. 2. Hopf action on Koszul CY algebras Let H be an involutory CY Hopf algebra, A a p-Koszul graded CY algebra such that A is an H-module algebra. In =-=[16]-=-, the authors proved that A#H is CY if and only if the homological determinant (for the definition, see for e.g. [14]) of the Hopf action on A is trivial. In this section, we want to show that the res... |

2 |
The Calabi-Yau property of smash products
- Yu, Zhang
(Show Context)
Citation Context ...is the free abelian group generated by µ. A more general result has been proved in [9] using a different method. Other results on the CY property of smash products can be found, e.g. in [7], [24] and =-=[26]-=-. In Section 3, we study the CY property of PBW deformations. The motivating examples come from pointed Hopf algebras of finite Cartan type. In [27], the authors showed that the liftings of a CY grade... |

1 | PBW deformations of Koszul algebras over a nonsemisimple ring, arXiv: 1312.2456v2
- He, Oystaeyen, et al.
(Show Context)
Citation Context ...have been studied, e.g. in [8], [15], [20] etc.. Recently, Walton and Witherspoon [23] proved the PBW theorem for any smash product algebra obtained from a Hopf algebra acting on a Koszul algebra. In =-=[11]-=-, He, Van Oystaeyen and Zhang proved the PBW theorem for a generalized smash product algebra obtained from a braiding between a finite dimensional algebra and a Koszul algebra, generalizing the corres... |

1 |
Deformations of quadratic algebras with antisymmetric relations
- He, Zhang
(Show Context)
Citation Context ...ssification. 16E65, 16S37, 16S40, 16S80 . Key words and phrases. Koszul algebras, Calabi-Yau algebras, smash products, PBW deformation . 1 2 XIAOLAN YU, FRED VAN OYSTAEYEN, AND YINHUO ZHANG cf. [10], =-=[12]-=- [27]. The aim of this paper is to investigate the Calabi-Yau property of the PBW deformation of the algebra A#H, where A is a Koszul CY algebra and H is a CY Hopf algebra acting on A. To this aim, we... |