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...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...

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... 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...

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...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...

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... 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...

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...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...

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...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 ...

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...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...

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...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 ...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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..., (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...

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...⊗ 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...

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...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...

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...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...

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...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...

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... 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...

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...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...

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...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...

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...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...