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- W133768664 abstract "This thesis is concerned with the Weyl algebras and the generalized Weyl algebras,nwhich are a formal algebraic generalization of Weyl algebras.nnn I have studied three separate problems. The first is investigation of similarity ofnideals - that is, the relation I ~ J if and only if R/ I nR/ J - in the first Weylnalgebra. Similarity of ideals is related to the existence of Ore localizations. A numbernof methods to attack the problem, using tools such as localizations and projectivenresolutions, are canvassed, unfortunately with little success.nnn The second problem is part of the study of noncommutative analogues of classicalngeometric ideas. I investigate a noncommutative analogue of fibres in generalizednWeyl algebras, and in particular the question of whether such fibres are disjoint asnin the commutative case. As always in noncommutative geometry, the definition andnarguments are essentially categorical and homological. The results diverge notablynfrom results previously obtained in Ore extensions. In particular, fibres in a generalizednWeyl algebra are not necessarily disjoint. I speculate that fibres which are notndisjoint correspond to the same irreducible varieties superimposed in different orders.nnn The third problem is a generalization of the b-function. The b-function plays annessential role in algorithms for the Weyl algebras, including algorithms for polynomialnand rational solutions of differential equations and calculation of restriction andnhomomorphism. I have generalized the modern computational definition, which differsnto some extent from the classical definition. The generalization retains somenof the computationally important properties of the b-function. In particular, analoguesnof polynomial solutions to differential equations can often be computed if thenb-function is known. The generalization applies to a class of graded algebras whichnI have called eigenspace graded. The necessary summary and generalization of thentheory of filtrations and gradings is also presented.nn I have also generalized the b-function algorithm to a certain class of generalizednWeyl algebra. The algorithm requires the theory of Grobner bases for rings compatiblenwith semigroup rings. I have summarised and stated in proper generality resultsnwhich are rather untidily presented in the literature. The algorithm also requires annew filtered-graded Grobner basis transfer result.nn" @default.
- W133768664 created "2016-06-24" @default.
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- W133768664 date "2005-01-01" @default.
- W133768664 modified "2023-10-17" @default.
- W133768664 title "Some computational and geometric aspects of generalized Weyl algebras" @default.
- W133768664 hasPublicationYear "2005" @default.
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