Generalizations of the Image Conjecture and the Mathieu Conjecture
By Wenhua Zhao. 2009
72 citations recorded by the source corpus.
We first propose a generalization of the image conjecture [Z3] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture [Ma] from integrals of $G$-finite functions over reductive Lie groups $G$ to integrals of polynomials over open subsets of $\mathbb R^n$ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of Duistermaat and van der Kallen's theorem [DK] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly-introduced notion.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/53d822168f346fb0934eb6df14a43c60e9d73254
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Citations in the corpus
- Images of commuting differential operators of order one with constant leading coefficients (2009, background): “ard generalization of the IC to C[z−1,z] does not always hold. For a modified generalization of the IC and their connections with the Mathieu conjecture and the Duistermaat-van der Kallen theorem, see [Z4]. 1.2. Arrangement. In section 2, we assume that Ais a commutative algebra over a field K of char”
- Mathieu Subspaces of Associative Algebras (2010, background): “ft (resp., right)Mathieu subspace ofA, and every (two-sided) ideal of Ais a (two-sided) Mathieu subspace and hence, also a pre-two-sided Mathieu subspace of A. But the converse is not true (see [DK], [Z4], [EWZ1], [FPYZ], [WZ] for some examples of Mathieu subspaces which are not ideals). Therefore, ”