|Title||Non-commutative Computer Algebra for polynomial algebras: Groebner bases, applications and implementation|
|Author(s)|| Viktor Levandovskyy|
|Text||Non-commutative Computer Algebra for polynomial algebras: Groebner bases, applications and implementation|
|Abstract||Non-commutative polynomial algebras appear in a wide range of applications, from quantum groups and theoretical physics to linear differential and difference equations.|
In the thesis, we have developed a framework, unifying many important
algebras in the classes of $G$-- and $GR$--algebras and studied their ring--theoretic properties. Let $A$ be a $G$--algebra in $n$ variables. We establish necessary and sufficient conditions for $A$ to have a Poincar'e--Birkhoff--Witt (PBW) basis. Further on, we show that besides the existence of a PBW basis, $A$ shares some other properties with the commutative polynomial ring $K[x_1,ldots,x_n]$. In particular, $A$ is a Noetherian integral domain of Gel'fand--Kirillov dimension $n$. Both Krull and global homological dimension of $A$ are bounded by $n$; we provide examples of $G$--algebras where these inequalities are strict. Finally, we prove that $A$ is Auslander--regular and a Cohen--Macaulay algebra.
In order to perform symbolic computations with modules over $GR$--algebras, we generalize Gröbner bases theory, develop and respectively enhance new and existing algorithms. We unite the most fundamental algorithms in a suite of applications, called "Gröbner basics" in the literature. Furthermore, we discuss algorithms appearing in the non-commutative case only, among others two--sided Gr"obner bases for bimodules, annihilators of left modules and operations with opposite algebras.
An important role in Representation Theory is played by various subalgebras, like the center and the Gel'fand--Zetlin subalgebra. We discuss their properties and their relations to Gröbner bases, and briefly comment some aspects of their computation. We proceed with these subalgebras in the chapter devoted to the algorithmic study of morphisms between $GR$--algebras. We provide new results and algorithms for computing the preimage of a left ideal under a morphism of $GR$--algebras and show both merits and limitations of several methods that we propose. We use this technique for the computation of the kernel of a morphism, decomposition of a module into central characters and algebraic dependence of pairwise commuting elements. We give an algorithm for computing the set of one--dimensional representations of a $G$--algebra $A$, and prove, moreover, that if the set of finite dimensional representations of $A$ over a ground field $K$ is not empty, then the homological dimension of $A$ equals $n$.
All the algorithms are implemented in a kernel extension Plural of the computer algebra system Singular. We discuss the efficiency of computations and provide a comparison with other computer algebra systems. We propose a collection of benchmarks for testing the performance of algorithms; the comparison of timings shows that our implementation outperforms all of the modern systems with the combination of both broad functionality and fast implementation.
In the thesis, there are many new non-trivial examples, and also the solutions to various problems, arising in different fields of mathematics. All of them were obtained with the developed theory and the implementation in Plural, most of them are treated computationally in this thesis for the first time.
|Keywords||Computer Algebra , Non-commutative Algebra, Groebner-Basis, Elimination Methods, Preimage of an ideal under a morphism of algebras, Algebraic dependence of commuting elements, Computer Algebra System|