On the consistency of twisted generalized Weyl algebras
Authors:
Vyacheslav Futorny and Jonas T. Hartwig
Journal:
Proc. Amer. Math. Soc. 140 (2012), 33493363
MSC (2010):
Primary 16D30, 16S35; Secondary 16S85
Published electronically:
February 17, 2012
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Abstract: A twisted generalized Weyl algebra of degree depends on a base algebra , commuting automorphisms of , central elements of and on some additional scalar parameters. In a paper by Mazorchuk and Turowska, it is claimed that certain consistency conditions for and are sufficient for the algebra to be nontrivial. However, in this paper we give an example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra to map injectively into . In particular they are sufficient for the algebra to be nontrivial. We speculate that these consistency relations may play a role in other areas of mathematics, analogous to the role played by the YangBaxter equation in the theory of integrable systems.
 1.
M.F. Atiyah, I.G. Macdonald, Introduction to commutative algebra, Volume 361 of AddisonWesley series in mathematics, Westview Press, 1994.
 2.
V.
V. Bavula, Generalized Weyl algebras and their
representations, Algebra i Analiz 4 (1992),
no. 1, 75–97 (Russian); English transl., St. Petersburg Math.
J. 4 (1993), no. 1, 71–92. MR 1171955
(93h:16043)
 3.
Georgia
Benkart and Matthew
Ondrus, Whittaker modules for generalized Weyl
algebras, Represent. Theory 13 (2009), 141–164. MR 2497458
(2010e:16039), http://dx.doi.org/10.1090/S1088416509003471
 4.
George
M. Bergman, The diamond lemma for ring theory, Adv. in Math.
29 (1978), no. 2, 178–218. MR 506890
(81b:16001), http://dx.doi.org/10.1016/00018708(78)900105
 5.
César
Gómez, Martí
RuizAltaba, and Germán
Sierra, Quantum groups in twodimensional physics, Cambridge
Monographs on Mathematical Physics, Cambridge University Press, Cambridge,
1996. MR
1408901 (97j:81143)
 6.
Jonas
T. Hartwig, Locally finite simple weight modules over twisted
generalized Weyl algebras, J. Algebra 303 (2006),
no. 1, 42–76. MR 2253653
(2008e:16021), http://dx.doi.org/10.1016/j.jalgebra.2006.05.036
 7.
Jonas
T. Hartwig, Twisted generalized Weyl algebras, polynomial Cartan
matrices and Serretype relations, Comm. Algebra 38
(2010), no. 12, 4375–4389. MR 2764825
(2012d:16070), http://dx.doi.org/10.1080/00927870903366926
 8.
David
A. Jordan, Primitivity in skew Laurent polynomial rings and related
rings, Math. Z. 213 (1993), no. 3,
353–371. MR 1227487
(94k:16009), http://dx.doi.org/10.1007/BF03025725
 9.
V.
Mazorchuk, M.
Ponomarenko, and L.
Turowska, Some associative algebras related to 𝑈(𝔤)
and twisted generalized Weyl algebras, Math. Scand.
92 (2003), no. 1, 5–30. MR 1951444
(2004i:16036)
 10.
Volodymyr
Mazorchuk and Lyudmyla
Turowska, Simple weight modules over twisted generalized Weyl
algebras, Comm. Algebra 27 (1999), no. 6,
2613–2625. MR 1687329
(2000j:16041), http://dx.doi.org/10.1080/00927879908826584
 11.
Volodymyr
Mazorchuk and Lyudmyla
Turowska, ∗representations of twisted generalized Weyl
constructions, Algebr. Represent. Theory 5 (2002),
no. 2, 163–186. MR 1909549
(2005h:46096), http://dx.doi.org/10.1023/A:1015669525867
 12.
Constantin
Năstăsescu and Freddy
Van Oystaeyen, Methods of graded rings, Lecture Notes in
Mathematics, vol. 1836, SpringerVerlag, Berlin, 2004. MR 2046303
(2005d:16075)
 13.
Alexander
L. Rosenberg, Noncommutative algebraic geometry and representations
of quantized algebras, Mathematics and its Applications,
vol. 330, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1347919
(97b:14004)
 1.
 M.F. Atiyah, I.G. Macdonald, Introduction to commutative algebra, Volume 361 of AddisonWesley series in mathematics, Westview Press, 1994.
 2.
 V.V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz 4, No. 1 (1992) 7597; English transl. in St. Petersburg Math. J. 4 (1993) 7192. MR 1171955 (93h:16043)
 3.
 G. Benkart, M. Ondrus, Whittaker modules for generalized Weyl algebras, Represent. Theory 13 (2009) 141164. MR 2497458 (2010e:16039)
 4.
 G.M. Bergman, The diamond lemma for ring theory, Advances in Mathematics 29 (1978) 178218. MR 506890 (81b:16001)
 5.
 C. Gómez, M. RuizAltaba, G. Sierra, Quantum groups in twodimensional physics, Cambridge University Press, Cambridge, 1996. MR 1408901 (97j:81143)
 6.
 J.T. Hartwig, Locally finite simple weight modules over twisted generalized Weyl algebras, J. Algebra 303, No. 1 (2006) 4276. MR 2253653 (2008e:16021)
 7.
 J.T. Hartwig, Twisted generalized Weyl algebras, polynomial Cartan matrices and Serretype relations, Comm. Algebr. 38 (2010), 43754389. MR 2764825
 8.
 D.A. Jordan, Primitivity in skew Laurent polynomial rings and related rings, Math. Z. 213 (1993) 353371. MR 1227487 (94k:16009)
 9.
 V. Mazorchuk, M. Ponomarenko, L. Turowska, Some associative algebras related to and twisted generalized Weyl algebras, Math. Scand. 92 (2003) 530. MR 1951444 (2004i:16036)
 10.
 V. Mazorchuk, L. Turowska, Simple weight modules over twisted generalized Weyl algebras, Comm. Alg. 27, No. 6 (1999) 26132625. MR 1687329 (2000j:16041)
 11.
 V. Mazorchuk, L. Turowska, *Representations of twisted generalized Weyl constructions, Algebr. Represent. Theory 5, No. 2 (2002) 163186. MR 1909549 (2005h:46096)
 12.
 C. Nastasescu, F. Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, 1836. SpringerVerlag, Berlin, 2004. MR 2046303 (2005d:16075)
 13.
 A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, Kluwer, Dordrecht, 1995. MR 1347919 (97b:14004)
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Additional Information
Vyacheslav Futorny
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo SP, 05315970, Brazil
Email:
futorny@ime.usp.br
Jonas T. Hartwig
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305
Email:
jonas.hartwig@gmail.com
DOI:
http://dx.doi.org/10.1090/S000299392012111840
PII:
S 00029939(2012)111840
Received by editor(s):
March 22, 2011
Received by editor(s) in revised form:
April 7, 2011
Published electronically:
February 17, 2012
Communicated by:
Kailash C. Misra
Article copyright:
© Copyright 2012
American Mathematical Society
