Exotic elliptic algebras of dimension 4

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

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Infinite Hopf family of elliptic algebras and bosonization

Journal of Physics A General Physics ◽

10.1088/0305-4470/32/10/012 ◽

1999 ◽

Vol 32 (10) ◽

pp. 1951-1959 ◽

Cited By ~ 1

Author(s):

Bo-Yu Hou ◽

Liu Zhao ◽

Xiang-Mao Ding

Keyword(s):

Elliptic Algebras

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Sklyanin elliptic algebras

Functional Analysis and Its Applications ◽

10.1007/bf01079526 ◽

1990 ◽

Vol 23 (3) ◽

pp. 207-214 ◽

Cited By ~ 30

Author(s):

A. V. Odesskii ◽

B. L. Feigin

Keyword(s):

Elliptic Algebras

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Constructions of Sklyanin elliptic algebras and quantumR-Matrices

Functional Analysis and Its Applications ◽

10.1007/bf01768666 ◽

1993 ◽

Vol 27 (1) ◽

pp. 31-38 ◽

Cited By ~ 4

Author(s):

A. V. Odesskii ◽

B. L. Feigin

Keyword(s):

Elliptic Algebras

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Hilbert series of modules of GK-dimension two over elliptic algebras

Journal of Algebra ◽

10.1016/j.jalgebra.2006.11.024 ◽

2007 ◽

Vol 311 (2) ◽

pp. 635-664 ◽

Cited By ~ 1

Author(s):

Koen De Naeghel

Keyword(s):

Hilbert Series ◽

Gk Dimension ◽

Elliptic Algebras

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Holonomic Poisson Manifolds and Deformations of Elliptic Algebras

Geometry and Physics: Volume II ◽

10.1093/oso/9780198802020.003.0028 ◽

2018 ◽

pp. 681-704 ◽

Cited By ~ 1

Author(s):

Brent Pym ◽

Travis Schedler

Keyword(s):

Symplectic Form ◽

Vector Fields ◽

Structural Features ◽

Poisson Manifolds ◽

Deformation Spaces ◽

Hamiltonian Vector Fields ◽

Finite Dimensional ◽

Degeneracy Condition ◽

Deformation Complex ◽

Elliptic Algebras

This chapter introduces a natural non-degeneracy condition for Poisson structures, called holonomicity, which is closely related to the notion of a log symplectic form. Holonomic Poisson manifolds are privileged by the fact that their deformation spaces are as finite-dimensional as one could ever hope: the corresponding derived deformation complex is a perverse sheaf. The chapter develops some basic structural features of these manifolds, highlighting the role played by the divergence of Hamiltonian vector fields. As an application, it establishes the deformation invariance of certain families of Poisson manifolds defined by Feigin and Odesskii, along with the ‘elliptic algebras’ that quantize them.

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