Poincaré–Birkhoff–Witt deformations of Calabi–Yau algebras

Berger, Roland; Taillefer, Rachel

doi:10.4171/jncg/6

Cited by 47 publications

(124 citation statements)

References 19 publications

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“…For example, the enveloping algebra of any finite-dimensional Lie algebra g is Calabi-Yau if trace(ad g (x)) for all x ∈ g ([13], Lemma 4.1). PBW deformations of Calabi-Yau algebras were studied by Berger and Taillefer [7]. Their main result is that PBW deformations of Calabi-Yau algebras defined by quivers and potentials are again Calabi-Yau.…”

Section: Skew Calabi-yau Algebrasmentioning

confidence: 99%

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PBW deformations of Artin–Schelter regular algebras

Gaddis

2016

J. Algebra Appl.

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We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. It is shown that the homogenization, when it is a geometric algebra, contains a component whose points are in 1-1 correspondence with the simple modules of the deformation. We classify all PBW deformations of 2-dimensional regular algebras and give examples of 3-dimensional deformations. Other properties, such as the skew Calabi-Yau property and closure under tensor products, are considered.

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Section: Skew Calabi-yau Algebrasmentioning

confidence: 99%

“…The Hochschild dimension of A is known to coincide with the global dimension of A ( [7], Remark 2.8).…”

Section: Skew Calabi-yau Algebrasmentioning

confidence: 99%

PBW deformations of Artin–Schelter regular algebras

Gaddis

2016

J. Algebra Appl.

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“…, x n ] be the polynomial algebra in n variables. Then A is a 2-Kozsul algebra (see [31], Example 1.6), A is a skew PBW extension (see [16], Example 5), A is Calabi-Yau of dimension n (see [9], page 18) and therefore, AS -regular (see [6], Proposition 4.3).…”

Section: The Polynomial Algebramentioning

confidence: 99%

“…(ii) Roland Berger y Rachel Taillefer in Proposition 4.3 of [6] show than if A is a connected N-graded Calabi-Yau algebra then A is AS -regular algebra, and in Proposition 5.4 they prove that if A is AS -regular C-algebra of global dimension 3 (with polynomial growth), then A is Calabi-Yau if and only if A is of type A in the classification of Artin and Schelter given in [2].…”

Section: Relations Examples and Counterexamplesmentioning

confidence: 99%

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Algunas relaciones entre álgebras N-Koszul, Artin-Schelter regular y Calabi-Yau con extensiones PBW torcidas

Suárez

Lezama

Reyes

2015

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Some authors have studied relations between Artin-Schelter regular algebras, N-Koszul algebras and CalabiYau algebras (resp. skew Calabi-Yau) of dimension d. In this paper we want to show through examples and counterexamples some relations between these classes of algebras with skew PBW extensions. In addition, we also exhibit some examples of the preservation of these properties by Ore extensions.Key words: Skew PBW extensions, Calabi-Yau algebras, N-Koszul algebras, AS -regular algebras, Ore extensions. ResumenAlgunos autores han estudiado las relaciones entre las álgebras Artin-Schelter regular, las álgebras N -Koszul y las álgebras Calabi-Yau (resp. skew Calabi-Yau) de dimensión d . En este artículo queremos mostrar a través de ejemplos y contraejemplos algunos relaciones entre estas clases de álgebras y las extensiones PBW torcidas. Además, mostraremos algunos ejemplos de preservación de estas propiedades en las extensiones de Ore.

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Equivalence of a-maximization and volume minimization

Eager

2014

J. High Energ. Phys.

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The low energy effective theory on a stack of D3-branes at a Calabi-Yau singularity is an N = 1 quiver gauge theory. The AdS/CFT correspondence predicts that the strong coupling dynamics of the gauge theory is described by weakly coupled type IIB supergravity on AdS 5 × L 5 , where L 5 is a Sasaki-Einstein manifold. Recent results on Calabi-Yau algebras efficiently determine the Hilbert series of any superconformal quiver gauge theory. We use the Hilbert series to determine the volume of the horizon manifold in terms of the fields of the quiver gauge theory. One corollary of the AdS/CFT conjecture is that the volume of the horizon manifold L 5 is inversely proportional to the a-central charge of the gauge theory. By direct comparison of the volume determined from the Hilbert series and the a-central charge, this prediction is proved independently of the AdS/CFT conjecture.

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Poincaré–Birkhoff–Witt deformations of Calabi–Yau algebras

Cited by 47 publications

References 19 publications

PBW deformations of Artin–Schelter regular algebras

PBW deformations of Artin–Schelter regular algebras

Algunas relaciones entre álgebras N-Koszul, Artin-Schelter regular y Calabi-Yau con extensiones PBW torcidas

Equivalence of a-maximization and volume minimization

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