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2010 | 8 | 6 | 1132-1155
Tytuł artykułu

A generalization of Mathieu subspaces to modules of associative algebras

Autorzy
Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
We first propose a generalization of the notion of Mathieu subspaces of associative algebras $$ \mathcal{A} $$, which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to $$ \mathcal{A} $$-modules $$ \mathcal{M} $$. The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of $$ \mathcal{A} $$-modules $$ \mathcal{M} $$, where R is the base ring of $$ \mathcal{A} $$. We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.
Wydawca
Czasopismo
Rocznik
Tom
8
Numer
6
Strony
1132-1155
Opis fizyczny
Daty
wydano
2010-12-01
online
2010-10-30
Twórcy
autor
Bibliografia
  • [1] Adjamagbo P.K., van den Essen A., A proof of the equivalence of the Dixmier, Jacobian and Poisson Conjectures, Acta Math. Vietnam., 2007, 32(2–3), 205–214
  • [2] Bass H., Connell E.H., 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
  • [3] Belov-Kanel A., Kontsevich M., The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Mosc. Math. J., 2007, 7(2), 209–218, 349
  • [4] Dixmier J., Sur les algèbres de Weyl, Bull. Soc. Math. France, 1968, 96, 209–242
  • [5] van den Essen A., Polynomial Automorphisms and the Jacobian Conjecture, Progr. Math., 190, Birkhäuser, Basel, 2000
  • [6] van den Essen A., The amazing image conjecture, preprint available at http://arxiv.org/abs/1006.5801
  • [7] 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
  • [8] van den Essen A., Wright D., Zhao W., Images of locally finite derivations of polynomial algebras in two variables, preprint available at http://arxiv.org/abs/1004.0521
  • [9] van den Essen A., Wright D., Zhao W., On the image conjecture, preprint available at http://arxiv.org/abs/1008.3962
  • [10] van den Essen A., Zhao W., Mathieu subspaces of univariate polynomial algebras (in preparation)
  • [11] Francoise J.P., Pakovich F., Yomdin Y., Zhao W., Moment vanishing problem and positivity: some examples, Bull. Sci. Math. (in press), DOI:10.1016/j.bulsci.2010.06.002
  • [12] Keller O.-H., Ganze Cremona-Transformationen, Monatsh. Math. Phys., 1939, 47(1), 299–306 http://dx.doi.org/10.1007/BF01695502
  • [13] Mathieu O., Some conjectures about invariant theory and their Applications, In: Algèbre non commutative, groupes quantiques et invariants, Reims, 1995, Semin. Congr., 2, Soc. Math. France, Paris, 1997, 263–279
  • [14] Pakovich F., On rational functions orthogonal to all powers of a given rational function on a curve, preprint available at http://arxiv.org/abs/0910.2105
  • [15] Tsuchimoto Y., Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math., 2005, 42(2), 435–452
  • [16] 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
  • [17] Zhao W., A vanishing conjecture on differential operators with constant coefficients, Acta Math. Vietnam., 2007, 32(3), 259–286
  • [18] 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
  • [19] 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
  • [20] 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. (in press), preprint available at http://arxiv.org/abs/0907.3991
  • [21] Zhao W., Mathieu subspaces of associative algebras, preprint available at http://arxiv.org/abs/1005.4260
  • [22] Zhao W., Willems R., Analogue of the Duistermaat-van der Kallen theorem for group algebras (in preparation)
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-010-0068-6
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