Hostname: page-component-7479d7b7d-pfhbr Total loading time: 0 Render date: 2024-07-10T14:33:04.306Z Has data issue: false hasContentIssue false

MODULES OVER QUANTUM LAURENT POLYNOMIALS

Published online by Cambridge University Press:  19 March 2012

ASHISH GUPTA*
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia (email: agupta@hri.res.in)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the Gelfand–Kirillov dimension for modules over quantum Laurent polynomials is additive with respect to tensor products over the base field. We determine the Brookes–Groves invariant associated with a tensor product of modules. We study strongly holonomic modules and show that there are nonholonomic simple modules.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Aljadeff, E. and Ginosar, Y., ‘On the global dimension of multiplicative Weyl algebra’, Arch. Math. 62 (1994), 401407.CrossRefGoogle Scholar
[2]Artamonov, V. A., ‘Projective modules over quantum algebras of polynomials’, Sb. Math. 82 (1995), 261269.CrossRefGoogle Scholar
[3]Artamonov, V. A., ‘Quantum polynomial algebras’, J. Math. Sci. 87 (1997), 34413462.CrossRefGoogle Scholar
[4]Artamonov, V. A., ‘The quantum Serre problem’, Russian Math. Surveys 53 (1998), 657730.CrossRefGoogle Scholar
[5]Artamonov, V. A., ‘General quantum polynomials: irreducible modules and Morita equivalence’, Izv. Math. 63 (1999), 336.CrossRefGoogle Scholar
[6]Bavula, V. V., ‘Module structure of the tensor product of simple algebras of Krull dimension one’, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 4 (1996), 721.Google Scholar
[7]Bavula, V. and van Oystaeyen, F., ‘Simple holonomic modules over the second Weyl algebra A2’, Adv. Math. 150 (2000), 80116.CrossRefGoogle Scholar
[8]Bieri, R. and Strebel, R., ‘Valuations and finitely presented metabelian groups’, Proc. Lond. Math. Soc. 41 (1980), 439464.CrossRefGoogle Scholar
[9]Bjork, J. E., Rings of Differential Operators (North-Holland, Amsterdam, 1979).Google Scholar
[10]Brookes, C. J. B., ‘Crossed products and finitely presented groups’, J. Group Theory 3 (2000), 433444.CrossRefGoogle Scholar
[11]Brookes, C. J. B. and Groves, J. R. J.., ‘Representations of the quantum torus and applications to finitely presented groups’, Preprint, arXiv:0711.2466v2 [math.RT].Google Scholar
[12]Brookes, C. J. B. and Groves, J. R. J., ‘Modules over nilpotent group rings’, J. Lond. Math. Soc. 52 (1995), 467481.CrossRefGoogle Scholar
[13]Brookes, C. J. B. and Groves, J. R. J., ‘Modules over crossed products of a division ring with an abelian group I’, J. Algebra 229 (2000), 2554.CrossRefGoogle Scholar
[14]Brookes, C. J. B. and Groves, J. R. J., ‘Modules over crossed products of a division ring with an abelian group II’, J. Algebra 253 (2002), 417445.CrossRefGoogle Scholar
[15]Jategaonkar, V. A., ‘A multiplicative analogue of the Weyl algebra’, Comm. Algebra 12 (1984), 16691688.CrossRefGoogle Scholar
[16]Lorenz, M., ‘Group rings and division rings’, in: Methods in Ring Theory (D. Reidel, Dordrecht, 1984), pp. 265280.CrossRefGoogle Scholar
[17]Manin, Y., Topics in Noncommutative Geometry (Princeton University Press, Princeton, NJ, 1991).CrossRefGoogle Scholar
[18]McConnell, J. C. and Pettit, J. J., ‘Crossed products and multiplicative analogues of Weyl algebras’, J. Lond. Math. Soc. 38 (1988), 4755.CrossRefGoogle Scholar
[19]Passman, D. S., Infinite Crossed Products (Academic Press, Boston, 1989).Google Scholar
[20]Robinson, D. J. S., A Course in the Theory of Groups (Springer, New York, 1996).CrossRefGoogle Scholar
[21]Wadsley, S. J., ‘Polyhedrality of the Brookes–Groves invariant for the noncommutative torus’, J. Algebra 293 (2005), 543560.CrossRefGoogle Scholar
[22]Wadsley, S. J., ‘Homogeneity and rigidity of the Brookes–Groves invariant for the noncommutative torus’, J. Algebra 297 (2006), 417437.CrossRefGoogle Scholar