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Canonical Bases for Algebraic Computations

Nordbeck, Patrik LU (2001) In Doctoral theses in mathematical sciences
Abstract
This thesis deals with computational methods in algebra, mainly focusing on the concept of Gröbner and SAGBI bases in non-commutative algebras. The material has a natural division into two parts. The first part is a rather extensive treatment of the basic theory of Gröbner bases and SAGBI bases in the non-commutative polynomial ring. The second part is a collection of six papers.



In the first paper we investigate, for quotients of the non-commutative polynomial ring, a property that implies finiteness of Gröbner bases computation, and examine its connection with Noetherianity. We propose a Gröbner bases theory for factor algebras, of particular interest for one-sided ideals, and show a few applications, e.g. how to... (More)
This thesis deals with computational methods in algebra, mainly focusing on the concept of Gröbner and SAGBI bases in non-commutative algebras. The material has a natural division into two parts. The first part is a rather extensive treatment of the basic theory of Gröbner bases and SAGBI bases in the non-commutative polynomial ring. The second part is a collection of six papers.



In the first paper we investigate, for quotients of the non-commutative polynomial ring, a property that implies finiteness of Gröbner bases computation, and examine its connection with Noetherianity. We propose a Gröbner bases theory for factor algebras, of particular interest for one-sided ideals, and show a few applications, e.g. how to compute (one-sided) syzygy modules. The material of the third paper is in some sense related to the contents of this first paper; in the third paper, the theory of SAGBI bases is extended to factor algebras.



The second and fourth paper concerns composition of polynomials. In the first of those two papers, we give sufficient and necessary conditions on a set of polynomials to guarantee that the property of being a non-commutative Gröbner basis is preserved after composition by this set. The latter paper treats the same problem for SAGBI bases.



In the fifth paper we introduce the concept of bi-automaton algebras, generalizing the automaton algebras previously defined by Ufnarovski. A bi-automaton algebra is a quotient of the free algebra, defined by a binomial ideal admitting a Gröbner basis which can be encoded as a regular set; we call such a Gröbner basis regular. We give several examples of bi-automaton algebras, and show how automata connected to regular Gröbner bases can be used to perform reduction.



In the last paper we investigate various important properties of regular languages associated with quotients of the free associative algebra. We suggest a generalization of a graph for normal words introduced by Ufnarovski, applicable to testing Noetherian properties of automaton algebras. Finally we show an alternative way to compute the generators for the Jacobson radical of any automaton monomial algebra. (Less)
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author
opponent
  • Prof Apel, Joachim, University of Leipzig, Germany
organization
publishing date
type
Thesis
publication status
published
subject
keywords
gruppteori, algebra, algebraisk geometri, fältteori, Talteori, group theory, algebraic geometry, field theory, Matematik, Number Theory, regular languages, Mathematics, composition of polynomials, factor algebras, Gröbner bases, SAGBI bases
in
Doctoral theses in mathematical sciences
pages
194 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Matematikcentrum, sal C
defense date
2001-10-05 10:15
external identifiers
  • other:LUTFMA-1012-2001
ISSN
1404-0034
ISBN
91-628-4969-7
language
English
LU publication?
yes
id
e8baa4f0-981d-4415-bf7c-7f8ba50bdb43 (old id 41926)
date added to LUP
2007-08-02 08:40:25
date last changed
2018-05-29 12:12:43
@phdthesis{e8baa4f0-981d-4415-bf7c-7f8ba50bdb43,
  abstract     = {This thesis deals with computational methods in algebra, mainly focusing on the concept of Gröbner and SAGBI bases in non-commutative algebras. The material has a natural division into two parts. The first part is a rather extensive treatment of the basic theory of Gröbner bases and SAGBI bases in the non-commutative polynomial ring. The second part is a collection of six papers.<br/><br>
<br/><br>
In the first paper we investigate, for quotients of the non-commutative polynomial ring, a property that implies finiteness of Gröbner bases computation, and examine its connection with Noetherianity. We propose a Gröbner bases theory for factor algebras, of particular interest for one-sided ideals, and show a few applications, e.g. how to compute (one-sided) syzygy modules. The material of the third paper is in some sense related to the contents of this first paper; in the third paper, the theory of SAGBI bases is extended to factor algebras.<br/><br>
<br/><br>
The second and fourth paper concerns composition of polynomials. In the first of those two papers, we give sufficient and necessary conditions on a set of polynomials to guarantee that the property of being a non-commutative Gröbner basis is preserved after composition by this set. The latter paper treats the same problem for SAGBI bases.<br/><br>
<br/><br>
In the fifth paper we introduce the concept of bi-automaton algebras, generalizing the automaton algebras previously defined by Ufnarovski. A bi-automaton algebra is a quotient of the free algebra, defined by a binomial ideal admitting a Gröbner basis which can be encoded as a regular set; we call such a Gröbner basis regular. We give several examples of bi-automaton algebras, and show how automata connected to regular Gröbner bases can be used to perform reduction.<br/><br>
<br/><br>
In the last paper we investigate various important properties of regular languages associated with quotients of the free associative algebra. We suggest a generalization of a graph for normal words introduced by Ufnarovski, applicable to testing Noetherian properties of automaton algebras. Finally we show an alternative way to compute the generators for the Jacobson radical of any automaton monomial algebra.},
  author       = {Nordbeck, Patrik},
  isbn         = {91-628-4969-7},
  issn         = {1404-0034},
  keyword      = {gruppteori,algebra,algebraisk geometri,fältteori,Talteori,group theory,algebraic geometry,field theory,Matematik,Number Theory,regular languages,Mathematics,composition of polynomials,factor algebras,Gröbner bases,SAGBI bases},
  language     = {eng},
  pages        = {194},
  publisher    = {Centre for Mathematical Sciences, Lund University},
  school       = {Lund University},
  series       = {Doctoral theses in mathematical sciences},
  title        = {Canonical Bases for Algebraic Computations},
  year         = {2001},
}