Year 2014,
Volume: 15 Issue: 15, 196 - 207, 01.06.2014
Abstract
In this paper, we consider centralizers of single elements in certain
Ore extensions, with a non-invertible endomorphism, of the ring of polynomials
in one variable over a field. We show that they are commutative and finitely
generated as algebras. We also show that for certain classes of elements their
centralizer is singly generated as an algebra.
Keywords
References
- S. A. Amitsur, Commutative linear differential operators, Pacific J. Math. 8 (1958), 1–10.
- D. Arnal and G. Pinczon, On algebraically irreducible representations of the Lie algebra sl(2), J. Math. Phys., 15 (1974), 350–359.
- V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz, 4(1) (1992), 75–97; translation in: St. Petersburg Math. J., 4(1), –92.
- J. P. Bell and L. W. Small, Centralizers in domains of Gelfand-Kirillov dimen- sion 2, Bull. Lond. Math. Soc., 36(6) (2004), 779–785.
- R. C. Carlson and K. R. Goodearl, Commutants of ordinary differential oper- ators, J. Differential Equations, 35(3) (1980), 339–365.
- J. Dixmier, Sur les alg`ebres de Weyl, Bull. Soc. Math. France, 96 (1968), 209–
- K. R. Goodearl, Centralizers in differential, pseudodifferential, and fractional differential operator rings, Rocky Mountain J. Math., 13(4) (1983), 573–618.
- K. R. Goodearl and R. B. Warfield, An introduction to noncommutative Noe- therian rings, second ed., London Mathematical Society Student Texts, vol. 61, Cambridge University Press, Cambridge, 2004.
- L. Hellstr¨om and S. D. Silvestrov, Ergodipotent maps and commutativity of elements in noncommutative rings and algebras with twisted intertwining, J. Algebra, 314(1) (2007), 17–41.
- L. Makar-Limanov, Centralizers in the quantum plane algebra, Studies in Lie theory, Progr. Math., vol. 243, Birkh¨auser Boston, Boston, MA, 2006, pp. 411–
- J. Richter, Burchnall-Chaundy theory for Ore extensions, Proceedings of the AGMP, Springer-Verlag, (to appear). X. Tang, Maximal commutative subalgebras of certain skew polynomial rings, Johan Richter Centre for Mathematical Sciences Lund University Box 118, SE-22199 Lund, Sweden e-mail: johanr@maths.lth.se Sergei Silvestrov
- Division of Applied Mathematics The School of Education, Culture and Communication M¨alardalen University Box 883, SE-72123 V¨aster˚as, Sweden e-mail: sergei.silvestrov@mdh.se
Year 2014,
Volume: 15 Issue: 15, 196 - 207, 01.06.2014
Abstract
References
- S. A. Amitsur, Commutative linear differential operators, Pacific J. Math. 8 (1958), 1–10.
- D. Arnal and G. Pinczon, On algebraically irreducible representations of the Lie algebra sl(2), J. Math. Phys., 15 (1974), 350–359.
- V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz, 4(1) (1992), 75–97; translation in: St. Petersburg Math. J., 4(1), –92.
- J. P. Bell and L. W. Small, Centralizers in domains of Gelfand-Kirillov dimen- sion 2, Bull. Lond. Math. Soc., 36(6) (2004), 779–785.
- R. C. Carlson and K. R. Goodearl, Commutants of ordinary differential oper- ators, J. Differential Equations, 35(3) (1980), 339–365.
- J. Dixmier, Sur les alg`ebres de Weyl, Bull. Soc. Math. France, 96 (1968), 209–
- K. R. Goodearl, Centralizers in differential, pseudodifferential, and fractional differential operator rings, Rocky Mountain J. Math., 13(4) (1983), 573–618.
- K. R. Goodearl and R. B. Warfield, An introduction to noncommutative Noe- therian rings, second ed., London Mathematical Society Student Texts, vol. 61, Cambridge University Press, Cambridge, 2004.
- L. Hellstr¨om and S. D. Silvestrov, Ergodipotent maps and commutativity of elements in noncommutative rings and algebras with twisted intertwining, J. Algebra, 314(1) (2007), 17–41.
- L. Makar-Limanov, Centralizers in the quantum plane algebra, Studies in Lie theory, Progr. Math., vol. 243, Birkh¨auser Boston, Boston, MA, 2006, pp. 411–
- J. Richter, Burchnall-Chaundy theory for Ore extensions, Proceedings of the AGMP, Springer-Verlag, (to appear). X. Tang, Maximal commutative subalgebras of certain skew polynomial rings, Johan Richter Centre for Mathematical Sciences Lund University Box 118, SE-22199 Lund, Sweden e-mail: johanr@maths.lth.se Sergei Silvestrov
- Division of Applied Mathematics The School of Education, Culture and Communication M¨alardalen University Box 883, SE-72123 V¨aster˚as, Sweden e-mail: sergei.silvestrov@mdh.se
There are 12 citations in total.
Details
| Other ID | JA36FT97AG |
|---|---|
| Authors | |
| Publication Date | June 1, 2014 |
| DOI | https://doi.org/10.24330/ieja.266247 |
| IZ | https://izlik.org/JA32PY86FE |
| Published in Issue | Year 2014 Volume: 15 Issue: 15 |