Noncommutative geometry based on commutator expansions

Noncommutative Geometry Based on Commutator Expansions

Lectures on Noncommutative Geometry

These Lectures are based on a course on noncommutative geometry given by the author in 2003. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results such as noncommutative Chern-Weil theory, noncommutative symplectic geometry, noncommutative diffe...

Notes on Noncommutative Geometry

Foreword This book is not just a survey of noncommutative geometry — later NCG; rather, it strives to answer some naive but vital questions: What is the purpose of NCG? What is it good for? Can NCG solve open problems of classical geometry inaccessible otherwise? In other words, why does NCG matter? What is it anyway? Good answer means good examples. A sweetheart of NCG called non-commutative t...

Notes on Noncommutative Geometry

Noncommutative geometry has roots in and is a synthesis of a number of diverse areas of mathematics, including: • Hilbert space and single operator theory; • Operator algebras (C*-algebras and von Neumann algebras); • Spin geometry – Dirac operators – index theory; • Algebraic topology – homological algebra. It has certainly also been inspired by quantum mechanics, and, besides feedback to the ...

Noncommutative Gröbner Bases for the Commutator Ideal

Let I denote the commutator ideal in the free associative algebra on m variables over an arbitrary field. In this article we prove there are exactly m! finite Gröbner bases for I , and uncountably many infinite Gröbner bases for I with respect to total division orderings. In addition, for m = 3 we give a complete description of its universal Gröbner basis. Let A be a finite set and let K be a f...