An analogue of the Duistermaat-van der Kallen theorem for group algebras

Wenhua Zhao; Roel Willems

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 974-986
  • ISSN: 2391-5455

Abstract

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Let G be a group, R an integral domain, and V G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen theorem [Duistermaat J.J., van der Kallen W., Constant terms in powers of a Laurent polynomial, Indag. Math., 1998, 9(2), 221–231], and also by recent studies on the notion of Mathieu subspaces, we show that for finite groups G, V G also forms a Mathieu subspace of the group algebra R[G] when certain conditions on the base ring R are met. We also show that for the free abelian groups G = ℤn, n ≥ 1, and any integral domain R of positive characteristic, V G fails to be a Mathieu subspace of R[G], which is equivalent to saying that the Duistermaat-van der Kallen theorem cannot be generalized to any field or integral domain of positive characteristic.

How to cite

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Wenhua Zhao, and Roel Willems. "An analogue of the Duistermaat-van der Kallen theorem for group algebras." Open Mathematics 10.3 (2012): 974-986. <http://eudml.org/doc/269550>.

@article{WenhuaZhao2012,
abstract = {Let G be a group, R an integral domain, and V G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen theorem [Duistermaat J.J., van der Kallen W., Constant terms in powers of a Laurent polynomial, Indag. Math., 1998, 9(2), 221–231], and also by recent studies on the notion of Mathieu subspaces, we show that for finite groups G, V G also forms a Mathieu subspace of the group algebra R[G] when certain conditions on the base ring R are met. We also show that for the free abelian groups G = ℤn, n ≥ 1, and any integral domain R of positive characteristic, V G fails to be a Mathieu subspace of R[G], which is equivalent to saying that the Duistermaat-van der Kallen theorem cannot be generalized to any field or integral domain of positive characteristic.},
author = {Wenhua Zhao, Roel Willems},
journal = {Open Mathematics},
keywords = {The Duistermaat-van der Kallen Theorem char 180 Mathieu subspaces char 180 Groups algebras; Mathieu subspaces; group algebras; finite groups; Duistermaat-van der Kallen theorem},
language = {eng},
number = {3},
pages = {974-986},
title = {An analogue of the Duistermaat-van der Kallen theorem for group algebras},
url = {http://eudml.org/doc/269550},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Wenhua Zhao
AU - Roel Willems
TI - An analogue of the Duistermaat-van der Kallen theorem for group algebras
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 974
EP - 986
AB - Let G be a group, R an integral domain, and V G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen theorem [Duistermaat J.J., van der Kallen W., Constant terms in powers of a Laurent polynomial, Indag. Math., 1998, 9(2), 221–231], and also by recent studies on the notion of Mathieu subspaces, we show that for finite groups G, V G also forms a Mathieu subspace of the group algebra R[G] when certain conditions on the base ring R are met. We also show that for the free abelian groups G = ℤn, n ≥ 1, and any integral domain R of positive characteristic, V G fails to be a Mathieu subspace of R[G], which is equivalent to saying that the Duistermaat-van der Kallen theorem cannot be generalized to any field or integral domain of positive characteristic.
LA - eng
KW - The Duistermaat-van der Kallen Theorem char 180 Mathieu subspaces char 180 Groups algebras; Mathieu subspaces; group algebras; finite groups; Duistermaat-van der Kallen theorem
UR - http://eudml.org/doc/269550
ER -

References

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  1. [1] Bass H., Connell E., Wright D., The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc., 1982, 7(2), 287–330 http://dx.doi.org/10.1090/S0273-0979-1982-15032-7 Zbl0539.13012
  2. [2] Duistermaat J.J., van der Kallen W., Constant terms in powers of a Laurent polynomial, Indag. Math., 1998, 9(2), 221–231 http://dx.doi.org/10.1016/S0019-3577(98)80020-7 Zbl0916.22007
  3. [3] van den Essen A., Polynomial Automorphisms and the Jacobian Conjecture, Progr. Math., 190, Birkhäuser, Basel, 2000 Zbl0962.14037
  4. [4] van den Essen A., The amazing image conjecture, preprint available at http://arxiv.org/abs/1006.5801 Zbl1239.14052
  5. [5] van den Essen A., Willems R., Zhao W., Some results on the vanishing conjecture of differential operators with constant coefficients, preprint available at http://arxiv.org/abs/0903.1478 Zbl1317.33007
  6. [6] van den Essen A., Wright D., Zhao W., Images of locally finite derivations of polynomial algebras in two variables, J. Pure Appl. Algebra, 2011, 215(9), 2130–2134 http://dx.doi.org/10.1016/j.jpaa.2010.12.002 Zbl1229.13022
  7. [7] van den Essen A., Wright D., Zhao W., On the image conjecture, J. Algebra, 2011, 340, 211–224 http://dx.doi.org/10.1016/j.jalgebra.2011.04.036 Zbl1235.14057
  8. [8] van den Essen A., Zhao W., Mathieu subspaces of univariate polynomial algebras, preprint available at http://arxiv.org/abs/1012.2017 Zbl1279.13011
  9. [9] Francoise J.P., Pakovich F., Yomdin Y., Zhao W., Moment vanishing problem and positivity: some examples, Bull. Sci. Math., 2011, 135(1), 10–32 http://dx.doi.org/10.1016/j.bulsci.2010.06.002 Zbl1217.44008
  10. [10] Keller O.-H., Ganze Cremona-Transformationen, Monatsh. Math. Phys., 1939, 47(1), 299–306 http://dx.doi.org/10.1007/BF01695502 
  11. [11] Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279 Zbl0889.22008
  12. [12] Passman D.S., The Algebraic Structure of Group Rings, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York-London-Sydney, 1977 Zbl0368.16003
  13. [13] Zhao W., Hessian nilpotent polynomials and the Jacobian conjecture, Trans. Amer. Math. Soc., 2007, 359(1), 249–274 http://dx.doi.org/10.1090/S0002-9947-06-03898-0 Zbl1109.14041
  14. [14] Zhao W., A vanishing conjecture on differential operators with constant coefficients, Acta Math. Vietnam., 2007, 32(2–3), 259–286 Zbl1139.14303
  15. [15] Zhao W., Images of commuting differential operators of order one with constant leading coefficients, J. Algebra, 2010, 324(2), 231–247 http://dx.doi.org/10.1016/j.jalgebra.2010.04.022 Zbl1197.14064
  16. [16] Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216 http://dx.doi.org/10.1016/j.jpaa.2009.10.007 Zbl1205.33017
  17. [17] Zhao W., A generalization of Mathieu subspaces to modules of associative algebras, Cent. Eur. J. Math., 2010, 8(6), 1132–1155 http://dx.doi.org/10.2478/s11533-010-0068-6 Zbl1256.16013
  18. [18] Zhao W., New proofs for the Abhyankar-Gurjar inversion formula and the equivalence of the Jacobian conjecture and the vanishing conjecture, Proc. Amer. Math. Soc., 2011, 139(9), 3141–3154 http://dx.doi.org/10.1090/S0002-9939-2011-10744-5 Zbl1229.14042
  19. [19] Zhao W., Mathieu subspaces of associative algebras, J. Algebra, 2012, 350(2), 245–272 http://dx.doi.org/10.1016/j.jalgebra.2011.09.036 
  20. [20] http://en.wikipedia.org/wiki/Newton’s_identities 
  21. [21] http://en.wikipedia.org/wiki/Cayley-Hamilton_theorem 

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