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A generalization of Mathieu subspaces to modules of associative algebras

100%
Zhao W.
Open Mathematics
|
2010
|
tom 8
|
nr 6
1132-1155
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.
2
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Some properties of and open problems on Hessian nilpotent polynomials

100%
Zhao W.
Annales Polonici Mathematici
|
2008
|
tom 93
|
nr 2
135-162
EN
In the recent work [BE1], [Me], [Burgers] and [HNP], the well-known Jacobian conjecture ([BCW], [E]) has been reduced to a problem on HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix is nilpotent) and their (deformed) inversion pairs. In this paper, we prove several results on HN polynomials, their (deformed) inversion pairs as well as on the associated symmetric polynomial or formal maps. We also propose some open problems for further study.
3
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A deformation of commutative polynomial algebras in even numbers of variables

100%
Zhao W.
Open Mathematics
|
2010
|
tom 8
|
nr 1
73-97
EN
We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [8] is reduced to an open problem on this deformation of polynomial algebras.
4
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An analogue of the Duistermaat-van der Kallen theorem for group algebras

63%
Zhao W., Willems R.
Open Mathematics
|
2012
|
tom 10
|
nr 3
974-986
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.
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