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Transactions of the American Mathematical Society

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Exotic elliptic algebras


Authors: Alex Chirvasitu and S. Paul Smith
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 16E65, 16S38, 16T05, 16W50
DOI: https://doi.org/10.1090/tran/7341
Published electronically: May 17, 2018
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Abstract: The 4-dimensional Sklyanin algebras, over $ \mathbb{C}$, $ A(E,\tau )$, are constructed from an elliptic curve $ E$ and a translation automorphism $ \tau $ of $ E$. The Klein vierergruppe $ \Gamma $ acts as graded algebra automorphisms of $ A(E,\tau )$. There is also an action of $ \Gamma $ as automorphisms of the matrix algebra $ M_2(\mathbb{C})$ making it isomorphic to the regular representation. The main object of study in this paper is the invariant subalgebra $ \widetilde {A}:=\big (A(E,\tau ) \otimes M_2(\mathbb{C})\big )^{\Gamma }$. Like $ A(E,\tau )$, $ \widetilde {A}$ is noetherian, generated by 4 degree-one elements modulo six quadratic relations, Koszul, Artin-Schelter regular of global dimension 4, has the same Hilbert series as the polynomial ring on 4 variables, satisfies the $ \chi $ condition, and so on. These results are special cases of general results proved for a triple $ (A,T,H)$ consisting of a Hopf algebra $ H$, an (often graded) $ H$-comodule algebra $ A$, and an $ H$-torsor $ T$. Those general results involve transferring properties between $ A$, $ A \otimes T$, and $ (A \otimes T)^{\rm {co} H}$. We then investigate $ \widetilde {A}$ from the point of view of non-commutative projective geometry. We examine its point modules, line modules, and a certain quotient $ \widetilde {B}:=\widetilde {A}/(\Theta ,\Theta ')$ where $ \Theta $ and $ \Theta '$ are homogeneous central elements of degree two. In doing this we show that $ \widetilde {A}$ differs from $ A$ in interesting ways. For example, the point modules for $ A$ are parametrized by $ E$ and 4 more points, whereas $ \widetilde {A}$ has exactly 20 point modules. Although $ \widetilde {B}$ is not a twisted homogeneous coordinate ring in the sense of Artin and Van den Bergh, a certain quotient of the category of graded $ \widetilde {B}$-modules is equivalent to the category of quasi-coherent sheaves on the curve $ E/E[2]$ where $ E[2]$ is the 2-torsion subgroup. We construct line modules for $ \widetilde {A}$ that are parametrized by the disjoint union $ (E/\langle \xi _1\rangle ) \sqcup (E/\langle \xi _2\rangle ) \sqcup (E/\langle \xi _3\rangle )$ of the quotients of $ E$ by its three subgroups of order 2.


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Additional Information

Alex Chirvasitu
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Address at time of publication: Department of Mathematics, University at Buffalo, Buffalo, New York 14260
Email: achirvas@buffalo.edu

S. Paul Smith
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: smith@math.washington.edu

DOI: https://doi.org/10.1090/tran/7341
Keywords: Sklyanin algebras, comodule algebras, torsors, descent
Received by editor(s): October 15, 2015
Received by editor(s) in revised form: February 13, 2017
Published electronically: May 17, 2018
Additional Notes: The first author acknowledges support from NSF grant DMS-1565226.
Article copyright: © Copyright 2018 American Mathematical Society

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