Algebraic structures as seen on the Weyl algebra

Recently there has been a growing number of interactions between noncommutative algebra and theoretical physics, often via noncommutative geometry. The algebras appearing are usually described by generators and relations what does not provide too much information about their structural properties. A well-known example of this type is the first Weyl algebra, also known as the basic algebra of quantum mechanics because of its direct link to the Heisenberg uncertainty relation. The Weyl algebra is an interesting example also from the algebraic point of view, it is more complicated than a matrix ring, yet it enjoys many structural properties. The idea behind these lecture notes is to develop some algebraic structure theory based upon properties observed on the Weyl algebra. One of the main methods is the use of filtrations and associated graded rings which is the topic of Chapter 3. The contents deals with : the relation with Lie algebras via the Heisenberg Lie algebra, finitelness conditions on rings and modules, localizations and rings of fractions, the relatioin to rings of differential operators, homological dimensions and the Gelfand-Kirillov dimension, module theory and holonomic modules, simple Noetherian algebras and semisimple rings and modules, etc...