Artin–Schelter Regular Algebras of Global Dimension Three

Stephenson, Darin R.

doi:10.1006/jabr.1996.0207

Cited by 56 publications

(49 citation statements)

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“…These algebras, also known as Sklyanin algebras, have been intensively studied in the literature (see [AS], [ATV], [AV] and references therein). In particular, they were classified in D. Stephenson's Ph.D. thesis [St2] (see also [St3]). The best understood case is that of singularities of type E 6 , resp.…”

Section: Quantizing Del Pezzo Surfacesmentioning

confidence: 98%

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Etingof¹,

Ginzburg²

2010

J. Eur. Math. Soc.

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Abstract. The hypersurface in C 3 with an isolated quasi-homogeneous elliptic singularity of type E r , r = 6, 7, 8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type E r provides a semiuniversal Poisson deformation of that Poisson structure.We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra C[x 1 , x 2 , x 3 ] to a noncommutative algebra with generators x 1 , x 2 , x 3 and the following three relations labeled by cyclic parmutations (i, j, k) of (1, 2, 3):This gives a family of Calabi-Yau algebras A t ( ) parametrized by a complex number t ∈ C × and a triple = ( 1 , 2 , 3 ) of polynomials of specifically chosen degrees.Our quantization of the coordinate ring of a del Pezzo surface is provided by noncommutative algebras of the form A t ( )/ , where ⊂ A t ( ) stands for the ideal generated by a central element which generates the center of the algebra A t ( ) if is generic enough.

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Section: Quantizing Del Pezzo Surfacesmentioning

confidence: 98%

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Etingof¹,

Ginzburg²

2010

J. Eur. Math. Soc.

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“…The noncommutative projective planes (where n = 3) were classified in [7,8] for algebras generated in degree 1 and in [65] for the general case. Their classification relied heavily on geometric techniques which we will soon discuss.…”

Section: Noncommutative Spacesmentioning

confidence: 99%

“…In dimensions 2 and 3 there exist classifications of such algebras (see [3] and [7,8,65] It would be interesting to know if in the dimension 3 case any of the families in the classification are related by cocycle twists. If that were the case then this behaviour would be similar to that which we will uncover in §6.2 for Rogalski and…”

Section: Chaptermentioning

confidence: 99%

Cocycle twists of algebras

Davies

2016

Communications in Algebra

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An be a connected graded k-algebra over an algebraically closed field k (thus A 0 = k). Assume that a finite abelian group G, of order coprime to the characteristic of k, acts on A by graded automorphisms. In conjunction with a suitable cocycle this action can be used to twist the multiplication in A. We study this new structure and, in particular, we describe when properties like Artin-Schelter regularity are preserved by such a twist. We then apply these results to examples of Rogalski and Zhang.

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“…In this result, Artin, Tate and Van den Bergh [6] classified the algebras generated in degree one, while Stephenson [26,27] did the general case.…”

Section: Historical Backgroundmentioning

confidence: 99%