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

Wenhua Zhao 1  and Roel Willems 2
  • 1 Illinois State University
  • 2 Radboud University Nijmegen

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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [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

OPEN ACCESS

Journal + Issues

Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.

Search