Exotic elliptic algebras
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- by Alex Chirvasitu and S. Paul Smith PDF
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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 $(A \otimes T)^\textrm {{co} H}$. 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.References
- Michael Artin, Geometry of quantum planes, Azumaya algebras, actions, and modules (Bloomington, IN, 1990) Contemp. Math., vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 1–15. MR 1144023, DOI 10.1090/conm/124/1144023
- M. Artin and J. T. Stafford, Semiprime graded algebras of dimension two, J. Algebra 227 (2000), no. 1, 68–123. MR 1754226, DOI 10.1006/jabr.1999.8226
- M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 33–85. MR 1086882
- M. Artin, J. Tate, and M. Van den Bergh, Modules over regular algebras of dimension $3$, Invent. Math. 106 (1991), no. 2, 335–388. MR 1128218, DOI 10.1007/BF01243916
- M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 (1990), no. 2, 249–271. MR 1067406, DOI 10.1016/0021-8693(90)90269-T
- M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228–287. MR 1304753, DOI 10.1006/aima.1994.1087
- Julien Bichon, Hopf-Galois objects and cogroupoids, Rev. Un. Mat. Argentina 55 (2014), no. 2, 11–69. MR 3285340
- A. P. Davies, Cocyle twists of algebras, PhD thesis, University of Manchester, 2014. https://www.escholar.manchester.ac.uk/uk-ac-man-scw:229719. Retrieved 01-20-2014.
- Andrew Davies, Cocycle twists of algebras, Comm. Algebra 45 (2017), no. 3, 1347–1363. MR 3573384, DOI 10.1080/00927872.2016.1178271
- Andrew Davies, Cocycle twists of 4-dimensional Sklyanin algebras, J. Algebra 457 (2016), 323–360. MR 3490085, DOI 10.1016/j.jalgebra.2016.01.046
- Boris Feigin and Alexander Odesskii, A family of elliptic algebras, Internat. Math. Res. Notices 11 (1997), 531–539. MR 1448336, DOI 10.1155/S1073792897000354
- I. M. Gelfand and A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 5–19 (French). MR 207918
- Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
- H. F. Kreimer and P. M. Cook II, Galois theories and normal bases, J. Algebra 43 (1976), no. 1, 115–121. MR 424782, DOI 10.1016/0021-8693(76)90146-0
- Thierry Levasseur and S. Paul Smith, Modules over the $4$-dimensional Sklyanin algebra, Bull. Soc. Math. France 121 (1993), no. 1, 35–90 (English, with English and French summaries). MR 1207244
- Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
- Jiří Matoušek, Using the Borsuk-Ulam theorem, Universitext, Springer-Verlag, Berlin, 2003. Lectures on topological methods in combinatorics and geometry; Written in cooperation with Anders Björner and Günter M. Ziegler. MR 1988723
- Susan Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. MR 590245
- Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
- A. V. Odesskii, Introduction to the theory of elliptic algebras, http://data.imf.au.dk/conferences/FMOA05/ellect.pdf. Retrieved 09-18-2014.
- A. V. Odesskiĭ and B. L. Feĭgin, Sklyanin’s elliptic algebras, Funktsional. Anal. i Prilozhen. 23 (1989), no. 3, 45–54, 96 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 3, 207–214 (1990). MR 1026987, DOI 10.1007/BF01079526
- A. V. Odesskiĭ and B. L. Feĭgin, Constructions of elliptic Sklyanin algebras and of quantum $R$-matrices, Funktsional. Anal. i Prilozhen. 27 (1993), no. 1, 37–45 (Russian); English transl., Funct. Anal. Appl. 27 (1993), no. 1, 31–38. MR 1225909, DOI 10.1007/BF01768666
- Manuel Reyes, Daniel Rogalski, and James J. Zhang, Skew Calabi-Yau algebras and homological identities, Adv. Math. 264 (2014), 308–354. MR 3250287, DOI 10.1016/j.aim.2014.07.010
- Hans-Jürgen Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), no. 1-2, 167–195. Hopf algebras. MR 1098988, DOI 10.1007/BF02764619
- Brad Shelton and Michaela Vancliff, Some quantum $\mathbf P^3$s with one point, Comm. Algebra 27 (1999), no. 3, 1429–1443. MR 1669119, DOI 10.1080/00927879908826504
- Brad Shelton and Michaela Vancliff, Schemes of line modules. I, J. London Math. Soc. (2) 65 (2002), no. 3, 575–590. MR 1895734, DOI 10.1112/S0024610702003186
- E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation, Funktsional. Anal. i Prilozhen. 16 (1982), no. 4, 27–34, 96 (Russian). MR 684124
- S. P. Smith, The four-dimensional Sklyanin algebras, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part I (Antwerp, 1992), 1994, pp. 65–80. MR 1273836, DOI 10.1007/BF00962090
- S. Paul Smith, Corrigendum to “Maps between non-commutative spaces”[ MR2052602], Trans. Amer. Math. Soc. 368 (2016), no. 11, 8295–8302. MR 3546801, DOI 10.1090/tran/6908
- S. P. Smith and J. T. Stafford, Regularity of the four-dimensional Sklyanin algebra, Compositio Math. 83 (1992), no. 3, 259–289. MR 1175941
- S. Paul Smith and J. M. Staniszkis, Irreducible representations of the $4$-dimensional Sklyanin algebra at points of infinite order, J. Algebra 160 (1993), no. 1, 57–86. MR 1237078, DOI 10.1006/jabr.1993.1178
- S. Paul Smith and James J. Zhang, A remark on Gelfand-Kirillov dimension, Proc. Amer. Math. Soc. 126 (1998), no. 2, 349–352. MR 1415339, DOI 10.1090/S0002-9939-98-04074-X
- S. Paul Smith, Maps between non-commutative spaces, Trans. Amer. Math. Soc. 356 (2004), no. 7, 2927–2944. MR 2052602, DOI 10.1090/S0002-9947-03-03411-1
- J. T. Stafford and M. van den Bergh, Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 171–216. MR 1816070, DOI 10.1090/S0273-0979-01-00894-1
- W. A. Stein et al., Sage Mathematics Software (Version 6.4.1). The Sage Development Team, 2015, http://www.sagemath.org.
- Darin R. Stephenson and Michaela Vancliff, Some finite quantum $\Bbb P^3$s that are infinite modules over their centers, J. Algebra 297 (2006), no. 1, 208–215. MR 2206855, DOI 10.1016/j.jalgebra.2005.04.005
- Darin R. Stephenson and Michaela Vancliff, Constructing Clifford quantum $\Bbb P^3$s with finitely many points, J. Algebra 312 (2007), no. 1, 86–110. MR 2320448, DOI 10.1016/j.jalgebra.2007.02.015
- John Tate and Michel van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124 (1996), no. 1-3, 619–647. MR 1369430, DOI 10.1007/s002220050065
- K.-H. Ulbrich, Galois extensions as functors of comodules, Manuscripta Math. 59 (1987), no. 4, 391–397. MR 915993, DOI 10.1007/BF01170844
- M. Van den Bergh, An example with $20$ point modules, circulated privately, 1988.
- Michel Van den Bergh, Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc. 154 (2001), no. 734, x+140. MR 1846352, DOI 10.1090/memo/0734
- M. Vancliff, K. Van Rompay, and L. Willaert, Some quantum $\textbf {P}^3$s with finitely many points, Comm. Algebra 26 (1998), no. 4, 1193–1208. MR 1612220, DOI 10.1080/00927879808826193
- J P, Literaturberichte: Lehrbuch der Algebra, Monatsh. Math. Phys. 21 (1910), no. 1, A21–A23 (German). von Heinrich Weber, Prof. der Mathem. a. d. Universität Straßburg, 3. Band: Elliptische Funktionen und Algebraische Zahlen. 2. Auflage. Braunschweig (Friedr. Vieweg u. Sohn), 1908, XVI+733 Seiten. MR 1548048, DOI 10.1007/BF01693264
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
- MR Author ID: 868724
- Email: achirvas@buffalo.edu
- S. Paul Smith
- Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
- MR Author ID: 190554
- Email: smith@math.washington.edu
- 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.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 279-333
- MSC (2010): Primary 16E65, 16S38, 16T05, 16W50
- DOI: https://doi.org/10.1090/tran/7341
- MathSciNet review: 3885145