PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2012 | 10 | 3 | 974-986
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
3
Strony
974-986
Opis fizyczny
Daty
wydano
2012-06-01
online
2012-03-24
Twórcy
autor
autor
Bibliografia
  • [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
  • [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
  • [3] van den Essen A., Polynomial Automorphisms and the Jacobian Conjecture, Progr. Math., 190, Birkhäuser, Basel, 2000
  • [4] van den Essen A., The amazing image conjecture, preprint available at http://arxiv.org/abs/1006.5801
  • [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
  • [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
  • [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
  • [8] van den Essen A., Zhao W., Mathieu subspaces of univariate polynomial algebras, preprint available at http://arxiv.org/abs/1012.2017
  • [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
  • [10] Keller O.-H., Ganze Cremona-Transformationen, Monatsh. Math. Phys., 1939, 47(1), 299–306 http://dx.doi.org/10.1007/BF01695502
  • [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
  • [12] Passman D.S., The Algebraic Structure of Group Rings, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York-London-Sydney, 1977
  • [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
  • [14] Zhao W., A vanishing conjecture on differential operators with constant coefficients, Acta Math. Vietnam., 2007, 32(2–3), 259–286
  • [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
  • [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
  • [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
  • [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
  • [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] http://en.wikipedia.org/wiki/Newton’s_identities
  • [21] http://en.wikipedia.org/wiki/Cayley-Hamilton_theorem
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0028-4
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.