## 16Z05 Computational aspects of associative rings [See also 68W30]

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- Doctoral Thesis (2)
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#### Keywords

- Algebraic dependence of commuting elements (2)
- Algebraische Abhängigkeit der kommutierende Elementen (2)
- Computer Algebra System (2)
- Computeralgebra System (2)
- Computeralgebra (1)
- Das Urbild von Ideal unter einen Morphismus der Algebren (1)
- Eliminationsverfahren (1)
- Gröbner-Basis (1)
- Morphismus (1)
- Nichtkommutative Algebra (1)

This thesis is devoted to constructive module theory of polynomial
graded commutative algebras over a field.
It treats the theory of Groebner bases (GB), standard bases (SB) and syzygies as well as algorithms
and their implementations.
Graded commutative algebras naturally unify exterior and commutative polynomial algebras.
They are graded non-commutative, associative unital algebras over fields and may contain zero-divisors.
In this thesis
we try to make the most use out of _a priori_ knowledge about
their characteristic (super-commutative) structure
in developing direct symbolic methods, algorithms and implementations,
which are intrinsic to graded commutative algebras and practically efficient.
For our symbolic treatment we represent them as polynomial algebras
and redefine the product rule in order to allow super-commutative structures
and, in particular, to allow zero-divisors.
Using this representation we give a nice characterization
of a GB and an algorithm for its computation.
We can also tackle central localizations of graded commutative algebras by allowing commutative variables to be _local_,
generalizing Mora algorithm (in a similar fashion as G.M.Greuel and G.Pfister by allowing local or mixed monomial orderings)
and working with SBs.
In this general setting we prove a generalized Buchberger's criterion,
which shows that syzygies of leading terms play the utmost important role
in SB and syzygy module computations.
Furthermore, we develop a variation of the La Scala-Stillman free resolution algorithm,
which we can formulate particularly close to our implementation.
On the implementation side
we have further developed the Singular non-commutative subsystem Plural
in order to allow polynomial arithmetic
and more involved non-commutative basic Computer Algebra computations (e.g. S-polynomial, GB)
to be easily implementable for specific algebras.
At the moment graded commutative algebra-related algorithms
are implemented in this framework.
Benchmarks show that our new algorithms and implementation are practically efficient.
The developed framework has a lot of applications in various
branches of mathematics and theoretical physics.
They include computation of sheaf cohomology, coordinate-free verification of affine geometry
theorems and computation of cohomology rings of p-groups, which are partially described in this thesis.

In this thesis we present the implementation of libraries center.lib and perron.lib for the non-commutative extension Plural of the Computer Algebra System Singular. The library center.lib was designed for the computation of elements of the centralizer of a set of elements and the center of a non-commutative polynomial algebra. It also provides solutions to related problems. The library perron.lib contains a procedure for the computation of relations between a set of pairwise commuting polynomials. The thesis comprises the theory behind the libraries, aspects of the implementation and some applications of the developed algorithms. Moreover, we provide extensive benchmarks for the computation of elements of the center. Some of our examples were never computed before.

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 \(\mathbb{K}[x_1,\ldots,x_n]\). 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 Gröbner 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.