Multiparameter twisted Weyl algebras. (English) Zbl 1260.16022
A twisted generalized Weyl algebra is defined by a certain presentation (generalizing the one of the classical generalized Weyl algebra), given some base algebra \(R\), \(n\) commuting automorphisms of \(R\) and \(n\) central elements in \(R\). In the paper under review the authors define a family of twisted generalized Weyl algebras which they call ‘multiparameter twisted Weyl algebras’.
The first main theorem of the paper parameterizes simple quotients of multiparameter twisted Weyl algebras in terms of maximal ideals of certain Laurent polynomial rings. The second main theorem gives an explicit relation between four twisted generalized Weyl algebras, namely the multiparameter quantized Weyl algebra, Jordan’s localization, a specific multiparameter Weyl algebra that the authors define and its certain quotient.
The first main theorem of the paper parameterizes simple quotients of multiparameter twisted Weyl algebras in terms of maximal ideals of certain Laurent polynomial rings. The second main theorem gives an explicit relation between four twisted generalized Weyl algebras, namely the multiparameter quantized Weyl algebra, Jordan’s localization, a specific multiparameter Weyl algebra that the authors define and its certain quotient.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 16S32 | Rings of differential operators (associative algebraic aspects) |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
twisted Weyl algebras; generalized Weyl algebras; weight modules; simple rings; Whittaker modules; simple modules; presentationsReferences:
| [1] | Alev, J.; Dumas, F., Sur le corps des fractions de certaines algèbres quantiques, J. Algebra, 170, 229-265 (1994) · Zbl 0820.17015 |
| [2] | Batra, P.; Mazorchuk, V., Blocks and modules for Whittaker pairs, J. Pure Appl. Algebra, 215, 7, 1552-1568 (2011) · Zbl 1228.17008 |
| [3] | Bavula, V. V., Generalized Weyl algebras and their representations, Algebra i Analiz. Algebra i Analiz, St. Petersburg Math. J., 4, 1, 71-92 (1993), English transl.: · Zbl 0807.16027 |
| [4] | Bavula, V. V., Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules, CMS Conf. Proc., 14, 83-107 (1993) · Zbl 0806.17023 |
| [5] | G. Benkart, unpublished manuscript.; G. Benkart, unpublished manuscript. |
| [6] | Benkart, G.; Ondrus, M., Whittaker modules for generalized Weyl algebras, Represent. Theory, 13, 141-164 (2009) · Zbl 1251.16020 |
| [7] | Cohen, M.; Rowen, L. H., Group graded rings, Comm. Algebra, 11, 11, 1253-1270 (1983) · Zbl 0522.16001 |
| [8] | Drozd, Yu. A.; Futorny, V. M.; Ovsienko, S. A., Harish-Chandra subalgebras and Gelfand-Zetlin modules, (Finite-Dimensional Algebras and Related Topics. Finite-Dimensional Algebras and Related Topics, Ottawa, ON, 1992. Finite-Dimensional Algebras and Related Topics. Finite-Dimensional Algebras and Related Topics, Ottawa, ON, 1992, NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci., vol. 424 (1994)), 79-93 · Zbl 0812.17007 |
| [9] | V. Futorny, J.T. Hartwig, On the consistency of twisted generalized Weyl algebras, Proc. Amer. Math. Soc., in press, available at arXiv:1103.4374 [math.RA]; V. Futorny, J.T. Hartwig, On the consistency of twisted generalized Weyl algebras, Proc. Amer. Math. Soc., in press, available at arXiv:1103.4374 [math.RA] · Zbl 1281.16031 |
| [10] | Hartwig, J. T., Locally finite simple weight modules over twisted generalized Weyl algebras, J. Algebra, 303, 1, 42-76 (2006) · Zbl 1127.16019 |
| [11] | Hartwig, J. T., Twisted generalized Weyl algebras, polynomial Cartan matrices and Serre-type relations, Comm. Algebra, 38, 12, 4375-4389 (2010) · Zbl 1213.16015 |
| [12] | Hartwig, J. T.; Öinert, J., Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras · Zbl 1276.16019 |
| [13] | Hayashi, T., \(q\)-Analogues of Clifford and Weyl algebras-spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys., 127, 129-144 (1990) · Zbl 0701.17008 |
| [14] | Jordan, D. A., Primitivity in skew Laurent polynomial rings and related rings, Math. Z., 213, 353-371 (1993) · Zbl 0797.16037 |
| [15] | Jordan, D. A., A simple localization of the quantized Weyl algebra, J. Algebra, 174, 267-281 (1995) · Zbl 0833.16025 |
| [16] | Maltsiniotis, G., Groupes quantiques et structures différentielles, C. R. Acad. Sci. Paris Sér. I Math., 311, 12, 831-834 (1990) · Zbl 0728.17010 |
| [17] | Mazorchuk, V.; Ponomarenko, M.; Turowska, L., Some associative algebras related to \(U(g)\) and twisted generalized Weyl algebras, Math. Scand., 92, 5-30 (2003) · Zbl 1031.17009 |
| [18] | Mazorchuk, V.; Turowska, L., Simple weight modules over twisted generalized Weyl algebras, Comm. Algebra, 27, 6, 2613-2625 (1999) · Zbl 0949.17003 |
| [19] | Mazorchuk, V.; Turowska, L., ⁎-Representations of twisted generalized Weyl constructions, Algebr. Represent. Theory, 5, 2, 163-186 (2002) · Zbl 1192.17002 |
| [20] | Pusz, W.; Woronowicz, S. L., Twisted second quantization, Rep. Math. Phys., 27, 2, 231-257 (1989) · Zbl 0707.47039 |
| [21] | Rosenberg, A. L., Noncommutative Algebraic Geometry and Representations of Quantized Algebras (1995), Kluwer: Kluwer Dordrecht · Zbl 0839.16002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
