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A primer of algebraic D-modules

capaThis book is an elementary introduction to the theory of algebraic D-modules. Its only pre-requisite is a basic knowledge of algebra, including some module theory. A good reference for all the required background is P. M. Cohn's excellent `Algebra, vol. 1' (John Wiley and Sons, 1984). The book concentrates on the simplest of all examples of a ring of differential operators, the Weyl algebra. This algebra first appeared in the work of the pioneers of quantum theory as the algebra generated by the moment and position operators in quantum mechanics. It has recently found applications in areas ranging from the representation theory of Lie algebras to combinatorics. Besides describing all the main concepts and operations on D-modules in the special case of the Weyl algebra, the book includes several applications, notably to the Jacobian conjecture, differential equations and automatic proof of identities.  The book is published by Cambridge University Press.

Conteúdo  

  • The Weyl algebra
  • Ideal structure of the Weyl algebra
  • Rings of differential operators
  • Jacobian conjecture
  • Modules over the Weyl algebra.
  • Differential operators.
  • Graded and filtered modules.
  • Noetherian rings and module
  • Dimension and multiplicity.
  • Holonomic modules.
  • Characteristic varieties.
  • Tensor products.
  • External products.
  • Inverse Image.
  • Embeddings.
  • Direct images.
  • Kashiwara's theorem.
  • Preservation of holonomy.
  • Stability of differential equations.
  • Automatic proof of identities.