Noncommutative Polynomial Algebras of Solvable Type and Their Modules

Basic Constructive-Computational Theory and Methods

Huishi Li

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000471120
• Edition: 1
• Topic: Mathematik
• Subtopic: Algebra

Noncommutative Polynomial Algebras of Solvable Type and Their Modules is the first book to systematically introduce the basic constructive-computational theory and methods developed for investigating solvable polynomial algebras and their modules. In doing so, this book covers:

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• A constructive introduction to solvable polynomial algebras and Gröbner basis theory for left ideals of solvable polynomial algebras and submodules of free modules

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• The new filtered-graded techniques combined with the determination of the existence of graded monomial orderings

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• The elimination theory and methods (for left ideals and submodules of free modules) combining the Gröbner basis techniques with the use of Gelfand-Kirillov dimension, and the construction of different kinds of elimination orderings

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• The computational construction of finite free resolutions (including computation of syzygies, construction of different kinds of finite minimal free resolutions based on computation of different kinds of minimal generating sets), etc.

This book is perfectly suited to researchers and postgraduates researching noncommutative computational algebra and would also be an ideal resource for teaching an advanced lecture course.

Chapter 1

Solvable Polynomial Algebras

DOI: 10.1201/9781003213192-1

This chapter is devoted to

• a constructive introduction to solvable polynomial algebras (Section 1.1, Section 1.2, Section 1.3);
• a constructive introduction to the Gröbner basis theory for left ideals of solvable polynomial algebras (Section 1.4, Section 1.5);
• establishing the elimination theory and methods for left ideals of solvable polynomial algebras (Section 1.6).

For the sake of saving space, an algorithmic approach to computing finite Gröbner bases for left ideals of solvable polynomial algebras is postponed to Section 2.3, where we will show that for free (left) modules over solvable polynomial algebras, a noncommutative version of the Buchberger criterion holds true for determining whether a finite subset of a submodule is a Gröbnerbasis, and that a noncommutative version of the Buchberger algorithm does exist for computing finite Gröbner bases in free modules (so a left ideal is just viewed as a submodule of the algebra con...