The Geometry of Representations of 3-Dimensional Sklyanin Algebras

Laet, Kevin De; Bruyn, Lieven Le

doi:10.1007/s10468-014-9515-6

Cited by 10 publications

(22 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This fact was stated as a conjecture in the first version of the paper and a proof of it was given by Kevin DeLaet in . We provide an independent short proof in Section 6.5.…”

Section: Introductionmentioning

confidence: 80%

See 1 more Smart Citation

Poisson geometry of PI three‐dimensional Sklyanin algebras

Walton

Wang

Yakimov

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

We give the three‐dimensional Sklyanin algebras S that are module‐finite over their center Z, the structure of a Poisson Z‐order (in the sense of Brown–Gordon). We show that the induced Poisson bracket on Z is nonvanishing and is induced by an explicit potential. The double-struckZ3×double-struckk×‐orbits of symplectic cores of the Poisson structure are determined (where the group acts on S by algebra automorphisms). In turn, this is used to analyze the finite‐dimensional quotients of S by central annihilators: there are three distinct isomorphism classes of such quotients in the case (n,3)≠1 and two in the case (n,3)=1, where n is the order of the elliptic curve automorphism associated to S. The Azumaya locus of S is determined, extending results of Walton for the case (n,3)=1.

show abstract

“…This fact was stated as a conjecture in the first version of the paper and a proof of it was given by Kevin DeLaet in . We provide an independent short proof in Section 6.5.…”

Section: Introductionmentioning

confidence: 80%

“…Previously Poisson geometry was not utilized for these types of representation theoretic questions; cf. .…”

Section: Introductionmentioning

confidence: 99%

Poisson geometry of PI three‐dimensional Sklyanin algebras

Walton

Wang

Yakimov

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In the cases discussed in this section so far, the Jacobian algebra has only onedimensional simple representations. Quantized affine three-space at roots of unity behaves differently (see [7,14]): corresponding to the potential W = xyz − qxzy on Q 1 has simple modules of dimension r > 1 if and only if q is a primitive r-th root of unity. Moreover, the space of one-dimensional representations is independent of q provided q = 1.…”

Section: 41mentioning

confidence: 99%

“…To motivate Conjecture 3.4, we recall some basic facts about simple representations of the Sklyanin algebras [2,3,14,32]. Let r = |σ| be the order of the translation σ : E pt → E pt .…”

Section: The Conjecturesmentioning

confidence: 99%

Motivic Donaldson–Thomas invariants of some quantized threefolds

Cazzaniga¹,

Morrison²,

Pym³

et al. 2017

J. Noncommut. Geom.

View full text Add to dashboard Cite

This paper is motivated by the question of how motivic Donaldson-Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi-Yau threefolds, defined by quivers with homogeneous potentials. These families give deformation quantizations of affine three-space, the resolved conifold, and the resolution of the transversal An-singularity. It turns out that their invariants are generically constant, but jump at special values of the deformation parameter, such as roots of unity. The corresponding generating series are written in closed form, as plethystic exponentials of simple rational functions. While our results are limited by the standard dimensional reduction techniques that we employ, they nevertheless allow us to conjecture formulae for more interesting cases, such as the elliptic Sklyanin algebras.

show abstract

“…In [7], it was proved that there was a connection between the superpotential defining the Sklyanin algebra A −2τ (E) and the central element c 3 of degree 3 in A τ (E). This connection however can better be explained using Heisenberg invariant elements of V ⊗ V ⊗ V .…”

Section: Central Elements and Heisenberg Invariantsmentioning

confidence: 99%

Quotients of degenerate Sklyanin algebras

Laet

2017

J. Algebra Appl.

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper it is shown how the Heisenberg group of order 27 can be used to construct quotients of degenerate Sklyanin algebras. These quotients have properties similar to the classical Sklyanin case in the sense that they have the same Hilbert series, the same character series and a central element of degree 3. Regarding the central element of a 3-dimensional Sklyanin algebra, a better way to view this using Heisenberg-invariants is shown.

show abstract

The Geometry of Representations of 3-Dimensional Sklyanin Algebras

Cited by 10 publications

References 19 publications

Poisson geometry of PI three‐dimensional Sklyanin algebras

Poisson geometry of PI three‐dimensional Sklyanin algebras

Motivic Donaldson–Thomas invariants of some quantized threefolds

Quotients of degenerate Sklyanin algebras

Contact Info

Product

Resources

About