Calabi-Yau pointed Hopf algebras of finite Cartan type

Yu, Xiaolan; Zhang, Yinhuo

doi:10.4171/jncg/144

Cited by 6 publications

(8 citation statements)

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“…Chelma showed that the algebras U q (g) are CY algebras [11,Theorem 3.3.2]. The CY property of the algebras U (D, λ) were discussed in [39]. In this section we will show that the cleft objects of the algebras U (D, λ) are all twisted CY algebras.…”

Section: Cleft Objects Of U (D λ)mentioning

confidence: 84%

“…(iii)The algebra H is CY if and only if p i=1 χ β i = ε and S 2 is an inner automorphism.Proof. (i) This is Theorem 2.2 in[39].…”

mentioning

confidence: 80%

“…Remark 3.4. Theorem 2.3 in [39] showed that the Nakayama automorphism of the algebra U (D, λ) is the algebra automorphism ν defined by ν(x k ) = p i=1,i =j k χ β i (g k )x k , for all 1 k θ, and ν(g) = ( p i=1 χ β i )(g) for all g ∈ Γ, where each j k , 1 k θ, is the integer such that β j k = α k . Now we show that the algebra automorphisms µ and ν only differ by an inner automorphism.…”

Section: Cleft Objects Of U (D λ)mentioning

confidence: 99%

“…The Calabi-Yau (CY for short) property of the algebras U (D, λ) are discussed in [39]. CY algebras were introduced by Ginzburg [19] in 2006.…”

Section: Introductionmentioning

confidence: 99%

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Cleft extensions of Koszul twisted Calabi–Yau algebras

Oystaeyen

Zhang

2016

Isr. J. Math.

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Let H be a twisted Calabi-Yau (CY) algebra and σ a 2-cocycle on H. Let A be an N -Koszul twisted CY algebra such that A is a graded H σ -module algebra. We show that the cleft extension A#σH is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an N -Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of U (D, λ), a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi-Yau algebras.

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Section: Cleft Objects Of U (D λ)mentioning

confidence: 84%

“…(iii)The algebra H is CY if and only if p i=1 χ β i = ε and S 2 is an inner automorphism.Proof. (i) This is Theorem 2.2 in[39].…”

mentioning

confidence: 80%

Section: Cleft Objects Of U (D λ)mentioning

confidence: 99%

“…The Calabi-Yau (CY for short) property of the algebras U (D, λ) are discussed in [39]. CY algebras were introduced by Ginzburg [19] in 2006.…”

Section: Introductionmentioning

confidence: 99%

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Cleft extensions of Koszul twisted Calabi–Yau algebras

Oystaeyen

Zhang

2016

Isr. J. Math.

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“…For example, those Hopf algebras in the classification of Calabi-Yau pointed Hopf algebras of finite Cartan type in [23].…”

Section: Proposition 41 Suppose That a Is A Koszul As-regular Algebmentioning

confidence: 99%

Nakayama automorphisms of double Ore extensions of Koszul regular algebras

2016

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Let A be a Koszul Artin-Schelter regular algebra and σ an algebra homomorphism from A to M 2×2 (A). We compute the Nakayama automorphisms of a trimmed double Ore extension A P [y 1 , y 2 ; σ ] [introduced in Zhang and Zhang (J Pure Appl Algebra 212: 2008)]. Using a similar method, we also obtain the Nakayama automorphism of a skew polynomial extension A[t; θ ], where θ is a graded algebra automorphism of A. These lead to a characterization of the Calabi-Yau property of A P [y 1 , y 2 ; σ ], the skew Laurent extension A[t ±1 ; θ ] and A[y ±11 , y ±1 2 ; σ ] with σ a diagonal type.

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Skew Calabi–Yau algebras and homological identities

Reyes

Rogalski

Zhang

2014

Advances in Mathematics

118

166

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Abstract. A skew Calabi-Yau algebra is a generalization of a Calabi-Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities we prove show (i) how the Nakayama automorphism of a smash product algebra A#H is related to the Nakayama automorphisms of a graded skew Calabi-Yau algebra A and a finite-dimensional Hopf algebra H that acts on it; (ii) how the Nakayama automorphism of a graded twist of A is related to the Nakayama automorphism of A; and (iii) that the Nakayama automorphism of a skew Calabi-Yau algebra A has trivial homological determinant in case A is noetherian, connected graded, and Koszul. IntroductionWhile the Calabi-Yau property originated in geometry, it now has incarnations in the realm of algebra that seem to be of growing importance. Calabi-Yau triangulated categories were introduced by Kontsevich [Ko] in 1998. See [Ke] for an introductory survey about Calabi-Yau triangulated categories. Calabi-Yau algebras were introduced by Ginzburg [Gi] in 2006 as a noncommutative version of coordinate rings of Calabi-Yau varieties. Since the late 1990s, the study of CalabiYau categories and algebras has been related to a large number of other topics such as quivers with superpotentials, DG algebras, cluster algebras and categories, string theory and conformal field theory, noncommutative crepant resolutions, and Donaldson-Thomas invariants. Some fundamental questions in the area were answered by Van den Bergh recently in [VdB3].One known method for constructing a noncommutative Calabi-Yau algebra is to form the smash product A#H of a Calabi-Yau algebra A with a Calabi-Yau Hopf algebra H that acts nicely on A. This phenomenon has been studied quite broadly; for instance, see [BSW], [Fa, Section 3] We will employ the following notation. Let A be an algebra over a fixed commutative base field k. The unmarked tensor ⊗ always means ⊗ k . Let M be an A-bimodule, and let µ, ν be algebra automorphisms of A. Then µ M ν denotes the induced A-bimodule such that µ M ν = M as a k-space, and where Definition 0.1. Let A be an algebra over k.(a) A is called skew Calabi-Yau (or skew CY, for short) if (i) A is homologically smooth, that is, A has a projective resolution in the category A e -Mod that has finite length and such that each term in the projective resolution is finitely generated, and (ii) there is an integer d and an algebra automorphism µ of A such that Calabi-Yau and µ A is inner (or equivalently, µ A can be chosen to be the identity map after changing the generator of the bimodule 1 A µ ).There is some variation in the literature concerning the exact definition of (skew) CY algebras. Ginzburg included in his definition of a CY algebra [Gi, Definition 3.2.3] the condition that the A e -projective resolution of A is self-dual, but Van den Bergh has shown that this is satisfied automatically [VdB3, Appendix C]. We are also not the first to study skew CY algebras; the notion has been calle...

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Calabi-Yau pointed Hopf algebras of finite Cartan type

Cited by 6 publications

References 17 publications

Cleft extensions of Koszul twisted Calabi–Yau algebras

Cleft extensions of Koszul twisted Calabi–Yau algebras

Nakayama automorphisms of double Ore extensions of Koszul regular algebras

Skew Calabi–Yau algebras and homological identities

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